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I then tried $k=6$ and found something even more interesting: for $n=1,2,3,4$, the values of $E(6,n)$ are $$1,\ \frac53,\ \frac{19}9,\ \frac{65}{27},$$ which very strongly suggests that $E(6,n) = \dfrac{3^n-2^n}{3^{n-1}}.$
If so, then it seems inevitable that the general formula must be $\boxed{E(2k,n) = \dfrac{k^n - (k-1)^n}{k^{n-1}}}.$
I'm not prepared to put money on it, but I'm 99% sure that $$E(50,50) = \dfrac{25^{50} - 24^{50}}{25^{49}} = 25 - 24\Bigl(\frac{24}{25}\Bigr)^{\!49} \approx 21.75.$$
Now it's time for Sunday lunch, so I'll leave it to others to see if those results can be proved.
I can report that simulation gives the mean number off after 50 flips of the 50 switchs is 21.74 with a standard error ~0.034
(another exercise on my Python learning curve)
.
Last edited:
#### Opalg
##### MHB Oldtimer
Staff member
I now see how to prove the formula for $E(2k,n)$, and it's really quite easy. When the $(n+1)$th switch is flipped, the probability that it is already on is $\dfrac{E(2k,n)}{2k}$, in which case the flip turns it off, and the number of "on" switches is reduced by $1$. Likewise, the probability that the $(n+1)$th switch is off is $\dfrac{2k-E(2k,n)}{2k}$, in which case the flip turns it on, and the number of "on" swiches is increased by $1$. Therefore $$E(2k,n+1) = \frac{E(2k,n)}{2k}(E(2k,n)-1) + \frac{2k-E(2k,n)}{2k}(E(2k,n)+1) = \frac{(k-1)E(2k,n) + k}k.$$ Now assume as an inductive hypothesis that $E(2k,n) = \dfrac{k^n-(k-1)^n}{k^{n-1}}$. Then $$E(2k,n+1) = \frac{(k-1)E(2k,n) + k}k = \frac{(k-1)k^n-(k-1)^{n+1} + k^n}{k^n} = \frac{k^{n+1} - (k-1)^{n+1}}{k^n},$$ which completes the inductive step. The base case $E(2k,1) = 1$ is easy to check, so the result is proved. | {
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# $\frac{1}{2^n}\binom{n}{n}+\frac{1}{2^{n+1}}\binom{n+1}{n}+...+\frac{1}{2^{2n}}\binom{2n}{n}=1$: short proof?
The identity $\frac{1}{2^n}\binom{n}{n}+\frac{1}{2^{n+1}}\binom{n+1}{n}+...+\frac{1}{2^{2n}}\binom{2n}{n}=1$ arises from a question on probability in my textbook. A proof by induction on $n$, which exploits the fact that $\binom{a}{b}+\binom{a}{b+1}=\binom{a+1}{b+1}$, is straightforward but not enlightening.
Is it possible to find any very clever approaches? Via a combinatorial or probabilistic interpretation, for instance?
Suppose you are flipping a coin until you get $n+1$ heads or $n+1$ tails. For $k=0,\dots,n$, what is the probability that you are done after exactly $n+1+k$ flips?
If the last flip was a tails, this means that in the former $n+k$ flips there were exactly $n$ tails, and this happens with probability $\frac{1}{2}\cdot\frac{1}{2^{n+k}}\binom{n+k}{n}$ (the first $\frac{1}{2}$ is due to assuming the last flip is tails). Same for heads, so the probability of finishing after exactly $n+1+k$ flips is $\frac{1}{2^{n+k}}\binom{n+k}{n}$.
Now just note that the number of flips always ends up between $n+1$ and $n+1+n$ (by pigeonhole principle).
Updated Solution
Here's a neater solution!
\begin{align} \sum_{k=0}^n \frac 1{2^{n+k}}\binom {n+k}n &=\frac 1{2^{2n}}\sum_{k=0}^n \binom {k+n}k 2^{n-k}\\ &=\frac 1{2^{2n}}\sum_{k=0}^n \binom {k+n}k \sum_{j=0}^{n-k}\binom {n-k}j\\ &=\frac 1{2^{2n}}\sum_{l=0}^n \binom {2n-l}{n-l}\sum_{j=0}^l \binom lj &&(l=n-k)\\ &=\frac 1{2^{2n}}\sum_{j=0}^n \sum_{l=j}^n\binom {2n-l}n\binom lj\\ &=\frac 1{2^{2n}}\sum_{j=0}^n \binom {2n+1}{n+j+1} &&(*)\\ &=\frac 1{2^{2n}}\sum_{j=n+1}^{2n+1} \binom {2n+1}j\\ &=\frac 1{2^{2n}}\cdot \frac 12\sum_{j=0}^{2n+1}\binom {2n+1}j &&\text{(by symmetry)}\\ &=\frac 1{2^{2n+1}}\cdot 2^{2n+1}\\ &=1\;\;\color{red}{\blacksquare}\end{align}
$\qquad \qquad \quad ^*\displaystyle\scriptsize\text{using }\sum_{r} \binom {a-r}{c}\binom {b+r}{d}=\binom {a+b+1}{c+d+1}$
Solution posted earlier | {
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Solution posted earlier
Here's a direct algebraic proof without using induction.
\begin{align} \sum_{k=0}^n \binom {k+n}k x^k(x+y)^{n-k} &=\sum_{k=0}^n \binom {k+n}k x^k\sum_{j=0}^{n-k}\binom {n-k}Jy^jx^{n-k-j}\\ &=\sum_{k=0}^n \binom{k+n}k\sum_{j=0}^{n-k}\binom{n-k}jy^jx^{n-j}\\ &=\sum_{\ell=0}^n\binom{2n-\ell}{n-\ell}\sum_{j=0}^\ell\binom{\ell}jy^jx^{n-j} &&(\ell=n-k)\\ &=\sum_{j=0}^n\sum_{\ell=j}^n (-1)^{n-\ell}\binom{-n-1}{n-\ell}(-1)^{\ell-j}\binom{-j-1}{\ell-j}y^jx^{n-j}\\ &=\sum_{j=0}^n(-1)^{n-j}y^jx^{n-j}\sum_{\ell=j}^n\binom{-n-1}{n-\ell}\binom{-j-1}{\ell-j}\\ &=\sum_{j=0}^n(-1)^{n-j}y^jx^{n-j}\binom{-n-j-2}{n-j} &&\text{(Vandermonde)}\\ &=\sum_{j=0}^n(-1)^{n-j}y^jx^{n-j}\cdot (-1)^{n-j}\binom{2n+1}{n-j}\\ &=\sum_{j=0}^n \binom{2n+1}{n-j}y^jx^{n-j}\\ &=\sum_{i=0}^n \binom{2n+1}i x^i y^{n-i}\\ \text{Put }x=y=1:\hspace{4cm}\\ \sum_{k=0}^n\binom {k+n}k2^{n-k} &=\sum_{i=0}^m\binom{2n+1}i &&(i=n-j)\\ &=\frac 12\cdot 2^{2n+1} &&\text{(by symmetry)}\\ &=2^{2n}\\ \sum_{k=0}^n \binom {n+k}k 2^{-k}&=2^n\\ \sum_{k=0}^n\frac 1{2^{n+k}} \binom {n+k}n&=1\\ \end{align}
• @robjohn - Thank you - very kind of you! Sep 1, 2017 at 7:13
Here is an answer based upon generating functions. It is convenient to use the coefficient of operator $[z^n]$ to denote the coefficient of $z^n$ in a series. This way we can write e.g. \begin{align*} [z^k](1+z)^n=\binom{n}{k} \end{align*}
We obtain \begin{align*} \color{blue}{\sum_{k=0}^n\binom{n+k}{k}\frac{1}{2^{n+k}}} &=\sum_{k=0}^n\binom{2n-k}{n-k}\frac{1}{2^{2n-k}}\tag{1}\\ &=\sum_{k=0}^\infty[z^{n-k}](1+z)^{2n-k}\frac{1}{2^{2n-k}}\tag{2}\\ &=2^{-2n}[z^n](1+z)^{2n}\sum_{k=0}^\infty\left(\frac{2z}{1+z}\right)^{k}\tag{3}\\ &=2^{-2n}[z^n](1+z)^{2n}\cdot\frac{1}{1-\frac{2z}{1+z}}\tag{4}\\ &=2^{-2n}[z^n](1+z)^{2n+1}\cdot\frac{1}{1-z}\tag{5}\\ &=2^{-2n}\sum_{k=0}^n[z^k](1+z)^{2n+1}\tag{6}\\ &=2^{-2n}\sum_{k=0}^n\binom{2n+1}{k}\tag{7}\\ &=2^{-2n}\frac{1}{2}2^{2n+1}\tag{8}\\ &\color{blue}{=1} \end{align*}
Comment: | {
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Comment:
• In (1) we change the order of summation $k \rightarrow n-k$.
• In (2) we apply the coefficient of operator. We also set the limit to $\infty$ without changing anything since we are adding zeros only.
• In (3) we do a rearrangement and apply the formula $[z^{p-q}]A(z)=[z^p]z^qA(z)$.
• In (4) we apply the geometric series expansion.
• In (5) we do some simplifications.
• In (6) we do the Cauchy multiplication with the geometric series $\frac{1}{1-x}$ and restrict the upper limit of the sum with $n$ since other terms do not contribute to $[z^n]$.
• In (7) we select the coefficient of $z^k$.
• In (8) we apply the binomial theorem. | {
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# Hard Probability Questions | {
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knew when we started that this material is tricky, as demonstrated by veridical paradoxes like the Monty Hall problem, the Girl Named Florida, and so on. I have been doing probability and statistics for decades and now spend a lot of time explaining ideas to everyone from school students to the government Chief Scientist, and I still find some concepts tricky. Then they both turn left and walk for another four feet, and. Probability Worksheets and Printables. (Total 2 marks) 2. For more difficult questions, the child may be. If three balls are drawn from the vessel at random, what is the probability that the first ball is red, the second ball is blue, and the third ball is white?. For instance, if an experiment has 20 total possible outcomes and only 10 of them are successful, the probability of that problem is 50 percent. Questions have been categorized so you can pick your favorite category or challenge your friends to the latest trivia. There is no bias over the contestants decision so each door has a probability of 1/3 being chosen. What is the probability of the occurrence of a number that is odd or less than 5 when a fair die is rolled. Regardless of your level of education or your familiarity with probability theory; if you would like to learn more about this fascinating subject, you are welcome. Probability Cards (Intermediate) What is the probability of choosing a particular card from a deck? Requires basic knowledge of standard playing cards. You know from above that she has an 8/125 probability of winning immediately, a 27/125 probability of losing immediately, and therefore a (125-8-27)/125 = 90/125 probability of getting back to deuce one way or another. Learn and practice basic word and conditional probability aptitude questions with shortcuts, useful tips to solve easily in exams. As for the exact probability of getting a dice problem is something only privy to those over at GMAC. Calculation problems c. Here are two probability problems which are | {
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privy to those over at GMAC. Calculation problems c. Here are two probability problems which are more difficult than they look. Statistic and Probability: Exam Questions. How do you measure 45 minutes using two such sticks? Note that sticks are made of different material and the burning speed along different sections are different so you can't use the length of the burnt section to estimate time. Just remember that probability questions are the best for this (counting questions are pretty good, too). How to Solve Probability Problems. This test is one of the California Standards Tests administered as part of the Standardized Testing and Reporting (STAR) Program under policies set by the State Board of Education. Practice Quiz for Probability of Inheritance: No. A few of the general questions require prior knowledge of the following: o Conditional probability: p(x|y) o Knowing that the probability of two independent events can be found by multiplying their individual probabilities. Las Vegas discussion forum - Probability of streaks, page 1. Let’s start by creating an […]. The probability of an event is always a number between zero and 100%. The probability that a red or blue marble will be selected is 9/14. For the prime. -----This is question has a subtle contingency and requires two probability equations to solve. It should be clear that the format with the highest probability to pass is the most attractive format. In 4 years, Gita will be twice as old as Harvey. All lined up and 1st four people on the line lose. Check your answers seem right. This video shows examples of using probability trees to work out the overall probability of a series of events are shown. It is an area of mathematics with many diverse applications. Conditional probability is based upon an event A given an event B has already happened: this is written as P(A | B). If you're behind a web filter, please make sure that the domains *. Liwayway draws 2 balls out of the bag. This is a very simple | {
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please make sure that the domains *. Liwayway draws 2 balls out of the bag. This is a very simple probability question in a software interview. In Need Loan Shark case a printer is equipped with a drying system, front side type of money document also dry, the following living conditions possess begun to print out a. I wanted pupil to leave exam and not end up saying "Ohh I could have done that question!" two minutes later. The judge is his ex-wife Gretchen, who wants to show him some sympathy, but the law clearly calls for two shots to be taken at Henry from close range. Question: The probability of a car passing a certain intersection in a 20 minute windows is 0. If you test a first chip and it appears to be good chip, what is the probability that second chip is also good? And at least is this probability is greater than 50% or not?. What was the weather like on a particular date? This is my second most-asked question. The aptitude section of these tests include the following concepts: computing probability of events involving rolling dice, tossing coins, picking cards from a pack of cards, selecting numbers, mutually exclusive. CIE IGCSE Maths exam revision with questions & model answers for the topic Probability | Paper 2 | Hard. Questions To Ask Drug Rehab Centers (FCR), a leading addiction treatment center in the US, provides supervised medical detox and rehab programs to treat alcoholism, drug addiction and co-occurring mental health disorders such as PTSD, depression and anxiety. Hard (for me) Probability Question. And to be effective readers means asking the hard questions. There are 52 cards in a deck. Direct Payday Lenders Co. Some recently asked Jane Street Quantitative Trader interview questions were, "Basic probability questions on Conditional probability, expectation and weighted expectation. Thus the probability for the spinner to land in any designated section is 1/10. A PFD value of zero (0) means there is no probability of failure (i. 1 What is the | {
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section is 1/10. A PFD value of zero (0) means there is no probability of failure (i. 1 What is the probability of the selected key being able to open the store room? 2 What is the probability that the chosen key chain is from the third set and the key does not open the door? 3 What is the probability that the chosen key opens the door and it came from the first key chain?. The probability that a woman has all three risk factors, given that she has A and B, is 1 3. In a random experiment the outcomes are independent so the first two statements are incorrect. The papers for STEP 2 and STEP 3 each consist of 12 questions: 8 pure, 2 mechanics, and 2 statistics/probability. What is the probability that they are both girls? (The answer is not 1:2) Need More Help on How to Solve Probability Problems? Ask a teacher. As for the exact probability of getting a dice problem is something only privy to those over at GMAC. THE MOST difficult probability question (Originally Posted: 11/17/2011) since we're posting "hardest" probability questions, here's a problem that's actually difficult to solve. Probability revision and probability tree diagrams. A discrete probability distribution is a table (or a formula) listing all possible values that a discrete variable can take on, together with the associated probabilities. Re: 3 hard statistics problems Well he assumes that because otherwise he wouldn't get the answer that I postulated. Question a couple of and find out how many other answers you obtain. High School Statistics and Probability Worksheets. A response will appear in the window below the question to let you know if you are correct. Brain teaser game: Hard Logic Probability Puzzle. Then they both turn left and walk for another four feet, and. What is the probability that she gets a red ball and a blue ball?. The formula for conditional probability is P(A|B) = P(A and B)/P(B) In this case. Test out your automobile well before investing in its maintenance. Probability is one | {
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In this case. Test out your automobile well before investing in its maintenance. Probability is one of the trickiest (and equally fascinating) topics in the CAT quantitative aptitude section. Identify the random variable X. It would be a great help In a school, all girls play at least one of hockey and netball. Below are three excruciatingly difficult dice problems. Do you think you are the best math student in your class? If so then you must have a knack for tackling some problems believed to be unsolvable by your fellow c. The complement rule is especially useful in the case where it hard to compute the probability of an event, but it is relatively easy to compute the probability of "not" the event. Welcome to Perfect English Grammar! Welcome! I'm Seonaid and I hope you like the website. Please use all of our printables to make your day easier. For example, one way to partition S is to break into sets F and Fc, for any event F. All probability questions are the same. 4 Determine the probability of, or the odds for and against, an outcome in a situation. RAID rebuild failure chance calculator - created and maintained by magJ. HARD Probability Math questions!!! Please help fast!!!? Can someone please help? Please show your work! Thanks 1. An algorithm for testing prime numbers is trial testing, test whether whether the number is dividable by an integer from 2 to its square root. T his GMAT practice question is a Probability question and is a problem solving question. A boundary point is any of the 40 points on the edge of this region for which both coordinates are integers boundary points are indicated as purple in the diagram. And best of all they all (well, most!) come with answers. Prepare sixth graders for higher level math with in-depth, comprehensive, and fun worksheets that cover the four basic operations as well as algebraic equations, number theory, fractions, decimals, geometry, probability, critical thinking, and much more. The meaning (interpretation) of | {
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decimals, geometry, probability, critical thinking, and much more. The meaning (interpretation) of probability is the subject of theories of probability. That model is shown in Table 19. by Marco Taboga, PhD. 24-28 using a sample drawn from NORC's probability-based AmeriSpeak Panel, which is designed to be representative of the U. Logic Problems In the Court of Law I. Probability Questions and answers PDF with solutions. Solved examples with detailed answer description, explanation are given and it would be easy to understand. Instructions Use black ink or ball-point pen. Choice (1)The probability that the two chosen squares have a common side is 1/18 Correct answer Explanatory Answer Hard. How to Solve Probability Problems. The question is incomplete, that is, it doesn't provide information regarding with the level of probability that you are familiar with. Great for revision. A white urn contains 10. Hey guys; I was just wondering if anyone had any thouhgts on or examples of the hardest possible probability questions we can be asked in 2u. Probability(not same digits) = 1 - 10/100 = 90/100. probability problems, probability, probability examples, how to solve probability word problems, probability based on area, examples with step by step solutions and answers, How to use permutations and combinations to solve probability problems, How to find the probability of of simple events, multiple independent events, a union of two events. Here are two probability problems which are more difficult than they look. Linear Algebra, Multivariable Calculus, Probability and Statistics Cal Poly SLO Stat 321 Probability and Statistics for Engineers and Scientists. Year 10 Interactive Maths (Mathematics or Math) - Second Edition by G S Rehill. Work out the probability that it is from the 20th century. Summary: Maybe it’d be fine, that this thing Wonwoo feels every time he’s at the Midnight Express with a certain someone. Probability, P = Number of Favourable Outcome/Total Number of | {
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Express with a certain someone. Probability, P = Number of Favourable Outcome/Total Number of Outcomes = 12/52= 3/13. Learn more about ACT Academy. Probability Questions and answers PDF with solutions. If there is a connection between A and F or C and D, then just one of the lights on the left, and one of the lights on the right is enough to get light in this system. We repeatedly toss the coin and keep a running estimate of the empirical probability of heads. Probability revision and probability tree diagrams. Probability Quiz Online Test 2019: Now, follow this page, and you will get the answer to most of the questions like Probability Aptitude, Probability Aptitude Questions and Answers PDF format, Probability Exam Questions and Answers, Probability Questions and Answers, Hard Probability Questions, Probability Problems for Aptitude with Solutions, Probability Aptitude Formulas, etc. Probability Class 10 Extra Questions Maths Chapter 15. An n p r question. What is the probability that the survey will show a greater percentage of. 25, calculating the requested probability involves just making a simple normal probability calculation: Now converting the Y scores to standard normal Z scores, we get:. Below is a list of great ideas for potential science fair projects. Modal verbs of probability exercise 2; Past modals exercise 1 (could have, should have, would have) Click here to return to the main modals page. Solved examples with detailed answer description, explanation are given and it would be easy to understand - Page 2. Do you think you are the best math student in your class? If so then you must have a knack for tackling some problems believed to be unsolvable by your fellow c. The probability that player b is the last to hold the ball is 0, given that a passes the ball to him first. You can’t just multiply probabilities like that unless the failures are uncorrelated. In response to a comment by Christoper, I found this sleek RAID rebuild failure chance | {
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uncorrelated. In response to a comment by Christoper, I found this sleek RAID rebuild failure chance calculator. Question 3: There are 15 groups of poker players in a hall including a group from Seattle. Improve your math knowledge with free questions in "Experimental probability" and thousands of other math skills. Questions have been categorized so you can pick your favorite category or challenge your friends to the latest trivia. When you take the actual test, you will mark your answers on a separate machine-scorable answer sheet. Thus, there is only one element of randomness, and therefore only one element of probability to consider. Hey guys; I was just wondering if anyone had any thouhgts on or examples of the hardest possible probability questions we can be asked in 2u. Here you can assume that if a child is a girl, her name will be Lilia with probability $\alpha \ll 1$ independently from other children's names. Mathematics Practice Test Page 15 Use the graph to answer questions 56, 57 & 58 The graph shows the price paid and weight for bags of sugar bought at different shops. This question might be a little old to be ever asked again but it is a good warm up. The Bertrand paradox is a problem within the classical interpretation of probability theory. Question from very important topics are covered by NCERT Exemplar Class 9. Probability worksheets will help students to practice all of these skills with a chance of success! Most Popular Statistics Worksheets this Week. Probability Example 1. For example, if the chance of A happening is 50%, and the same for B, what are the chances of both happening, only one happening , at least one happening, or neither happening, and so on. Start by writing down the probability of a dice throw = 'lazy' (clue it will be a ratio less than 1) Then consider how you work out "probability of exactly n 6's when throwing S dice" (clue = it will involve 'factorials') PS the probability of you passing this course if some-one else does | {
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= it will involve 'factorials') PS the probability of you passing this course if some-one else does your homework is close to zero. I have tried to complete several of the sections, but I think I am getting lost somewhere. He wants to form 5 lines of 3 forwards each (left-wing, center,and right-wing). Johnsonbaugh said you would need at least N = 3,748,630 terms. Choice (1)The probability that the two chosen squares have a common side is 1/18 Correct answer Explanatory Answer Hard. You cannot develop a deep understanding and application of machine learning without it. By the way I was wondering whether it may be interesting to add to the Stata bookshelf a tutorial-based textbook on probability (let's say from basics to Bayesian probability distributions) issues related to biostatistics, epidemiology, social sciences and alike (with alike I mean all the. Statistic and Probability: Exam Questions. Fully worked-out solutions of these problems are also given, but of course you should first try to solve the problems on your own! c 2013 by Henk Tijms, Vrije University, Amsterdam. slide 1: In the x-y plane the square region bound by 00 10 0 10 10 and 0 10 is isolated. Treat These 15 Laptop Computers As A Random Sample And Use X To Denote The Number Of Them With Hard Disks Made By Western Digital In This Sample. All you need are two numbers to determine probability. Each face cube has probability 1/6 of being in the correct orientation. uk Probability 1 (H) - Version 2 January 2016 Probability 1 (H) A collection of 9-1 Maths GCSE Sample and Specimen questions from AQA, OCR, Pearson-Edexcel and WJEC Eduqas. Do you want to learn more about this topic? Then watch a video about finding the probability of simple independent events or two online videos about finding the probability of compound. Regardless of your level of education or your familiarity with probability theory; if you would like to learn more about this fascinating subject, you are welcome. How do we test if these | {
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would like to learn more about this fascinating subject, you are welcome. How do we test if these are the same? How do we estimate di erences between the probability of being eaten in di erent groups?. For any two of the three factors, the probability is 0. her coach has asked her to keep practicing until she scores 50 goals. Standard probability themes like coins, spinners, number cubes, and marbles are featured along with some other situations. Each face cube has probability 1/6 of being in the correct orientation. You roll a single die numbered from 1 to 6 twice. Logic Problems In the Court of Law I. A multiple-choice question on an economics quiz contains 10 questions with five possible answers each. In probability samples, each member of the population has a known non-zero probability of being selected. Understanding probability isn't just important for improving your odds at the roulette table in Las Vegas. Thank you all your hard work! You are my 'go to' person for insight as to what is happening back in Blighty. Probability model for one die. Probability: interpret graphs interpretation, mean, median and mode, simple probability calculations, data collection and analysis. The skills in this section have an unlimited potential in the business world. It is an area of mathematics with many diverse applications. The questions in the practice test in this book illustrate the types of multiple-choice questions in the test. Concepts questions b. See preview. There is no bias over the contestants decision so each door has a probability of 1/3 being chosen. Good luck!. Here are two probability problems which are more difficult than they look. These Data Scientist job interview questions will set the foundation for data science interviews to impress potential employers by knowing about your subject and being able to show the practical implications of data science. The sum of probabilities of all sample points in a sample space is equal to 1. Let your brain have some | {
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sum of probabilities of all sample points in a sample space is equal to 1. Let your brain have some fun with these brain games. But what can be said for the probability that at least 1 of 3 child births for this couple will have red hair? How can it truly be (7/8)^3 -1 When the carrier status of the unknown spouse is not a factor that gets reshuffled each time a new kid is born? That's my question. Click here to read the solution to this question Click here to return to the index. If someone could help me out with writing this out and also getting the actual probability of this scenario, I would really appreciate it! If you have any questions or need any more factors, just let me know but I think that what was given should be enough to generate an equation and get an answer. Probability deals with how likely it is that something will happen. A “bad beat” happens when a player completes a hand that started out with a very low probability of success. Studyclix makes exam revision and study easier. For any two of the three factors, the probability is 0. It quickly and easily calculates the probability of a URE failing during a RAID 5 or RAID 6 rebuild. a) If you randomly observe 200 cars, what is the mean?. Check your answers seem right. A measure of a player’s experience and maturity is how he handles bad beats. You are sure this will result in your best your score yet. The probability of getting 0-0-0 is 1 out of 1000, or 1 ÷1000 = 0. Learn vocabulary, terms, and more with flashcards, games, and other study tools. ALL answers should be fractions in simplest form. I like that. As in the Monty Hall problem, the intuitive answer is 1 / 2, but the probability is actually 2 / 3. Compute the probability of randomly guessing the answers and getting exactly 9 questions correct. Short Answers to Hard Questions About Zika Virus. This is 1/6. Find the probability that the total of the. Hard probability problem: How to calculate the the probability of having selected only < 70% | {
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of the. Hard probability problem: How to calculate the the probability of having selected only < 70% of all marbles in a bag, given the following draw rules? Ask Question Asked 1 year, 10 months ago. Conditional probability tells us the probabiity of event A given that B has occurred. Great Expectations: Probability Through Problems The resources found on this page offer a new approach to teaching probability. ξ = 120 coins in the collection T = coins from the 20th century B = British coins A coin is chosen at random. Conditional probability is based upon an event A given an event B has already happened: this is written as P(A | B). Probability of getting no head = P(all tails) = 1/32. A bag of jellybeans has 20 watermelon jellybeans, 45 sour apple jellybeans, 30 orange jellybeans and 5 cotton candy jellybeans. Liwayway draws 2 balls out of the bag. Wasn't sure how it would go or if they'd solve it. Probability is the likelihood of something happening or being true. This section provides the course exams with solutions and practice exams with solutions. Be wary of automobile professionals who. Can you please describe your steps. just been exchanged. For the Giants to win the series in game six, they need to win three games in five trials. Instructions Use black ink or ball-point pen. com is the place to go to get the answers you need and to ask the questions you want. Like dependability, this is also a probability value ranging from 0 to 1, inclusive. Then they both turn left and walk for another four feet, and. Probability Page 1 of 15 Probability Rules A sample space contains all the possible outcomes observed in a trial of an experiment, a survey, or some random phenomenon. Before being able to solve a probability question, you must first. The problems tend to be easy to state and understand, but sometimes irritatingly difficult to solve. 1 that a woman in the population has only this risk factor (and no others). There is no bias over the contestants decision so | {
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has only this risk factor (and no others). There is no bias over the contestants decision so each door has a probability of 1/3 being chosen. Released Test Questions Math 3 Introduction - Grade 3 Mathematics The following released test questions are taken from the Grade 3 Mathematics Standards Test. In fact, many probability questions are a set of two permutation probability questions with the denominator being the total number of outcomes for an event and the numerator being the number of favorable outcomes. Brain teaser game: Hard Logic Probability Puzzle. population. The probability of an event occurring is the chance or likelihood of it occurring. Model Question Paper Mathematics Class XII Time Allowed : 3 hours Max: Marks: 100 General Instructions (i) The question paper consists of three parts A, B and C. | {
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# When chessboards meet dominoes
You probably have heard about the following brainteaser :
Consider a $$8\times 8$$ chessboard. Remove two extreme squares (top-left and bottom-right e.g.). Can you fill the remaining chessboard with $$1\times 2$$ dominoes ?
The answer lies in a coloring argument. The problem, however, does not use the fact that the chessboard is colored. I would like to know improved examples of coloring problems like this one.
For instance, could you extend the problem with Tetris-like L-shaped dominoes and solve it using more than two colors ?
• Well, with 2x3 "dominos" you notice that 6 doesn't divide 64... Jul 30 '12 at 16:44
• 2x3 means L shape. Jul 30 '12 at 16:45
• Could you clarify: When you talk about using Tetris pieces, you are talking about an ordinary chessboard, not one with two pieces removed? Jul 30 '12 at 16:55
• @SiliconCelery : as you can add two L-shaped dominos to get a 2x4 rectangle, they can fit a regular chessboard. Concerning my question I guess the problem should be different from the beginning, it should probably substract each of the extreme squares to get 60 = 15 x 4 squares Jul 30 '12 at 17:01
• The 35 hexominoes cannot be packed into a rectangle as you have two more squares of one color than the other. See en.wikipedia.org/wiki/Hexomino. I have seen many math puzzles that exploit coloring. For trominoes you often color in strips of three colors or diagonally like a checkerboard with three colors. Jul 30 '12 at 17:59
Solomon Golomb's book Polyominoes presents a number of arguments of this type.
One that I remember is: a square is deleted from an 8×8 checkerboard. Can the remaining 63 squares be covered by 21 1×3 rectangles? The answer involves coloring the checkerboard in three colors in alternating diagonal stripes: | {
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This colors the 64 squares of the checkerboard with 21 green squares, 21 yellow squares, and 22 blue squares. Each 1×3 rectangle must cover exactly one square of each color. The deleted square therefore cannot be any of the green or yellow ones, nor any of the squares equivalent to one of these under a rotation or reflection of the checkerboard:
(This is four copies of the first diagram, superimposed, with suitable rotations.)
This eliminates all but 4 squares from consideration, namely the four bright blue ones in the previous diagram. So the only solutions involve deleting one of these four blue squares.
There are a number of analogous arguments about polyhexes that depend on a three-coloring of a hexagonal lattice:
For example, there are three different trihexes, which are made by joining three hexagons; two of these are guaranteed to cover exactly one cell of each color, no matter how they are placed.
I once wasted a lot of time trying to make myself a set of tetrominoes by marking up a 4x5 rectangle and cutting it apart, and I felt rather foolish when I realized that a straightforward checkerboard coloring shows that this is impossible. There are 5 tetrominoes, and four of them must cover two black and two white squares each. The 5th is T-shaped, and must cover three black squares and one white (or vice versa).
So they cannot possibly tile a 4×5 rectangle, which has equal numbers of black and white squares.
• Very nice example of coloring pattern. Jul 30 '12 at 19:38
A woodworm is sitting at the centre of a cube that's divided into $3^3$ identical cubelets. The woodworm can go from the centre of one cubelet to the centre of another in any edge-parallel direction. The woodworm would like to eat its way through the cube such that it visits the centre of each cubelet exactly once. Is this possible?
• Same principle, I see. Awesome ! Jul 30 '12 at 19:06 | {
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• Same principle, I see. Awesome ! Jul 30 '12 at 19:06
I searched for math.SE questions and answers that involve colouring of tilings and the like; here are two interesting ones that I found:
The Mathematics of Tetris
A tiling puzzle/question | {
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# Solve for upper/lower bound of a numerical definite integral
Hi, I’m trying to solve for b, given f, a and k, please help, Is there a way to do it in julia?
quadgk(f,a,b)=k
Thanks
2 Likes
There’s always a Newton iteration:
b = some_initial_guess
tol = sqrt(eps())
while true
δb = (quadgk(f,a,b, rtol=tol*0.1)[1] - k) / f(b)
b -= δb
abs(δb) ≤ tol * abs(b) && break
end
Note that I’m using the fact that \frac{d}{db} \int_a^b f(x) dx = f(b).
10 Likes
@stevengj, this is excellent.
Below a summary of different solutions, including yours and your advice to run Optim properly (even if not recommended for this problem kept it for sake of completeness).
f(x) = exp(x)*sin(x)
# forward modelling
a = 0
# b = π : is UNKNOWN to be found given f, a and k
k = 0.5*(1.0 + exp(1)^π) # k = 12.0703463163896
# SOLUTION-1: Newton solution by @stevengj
b = 1.0 # initial guess
tol = sqrt(eps())
while true
δb = (quadgk(f,a,b, rtol=tol*0.1)[1] - k) / f(b)
b -= δb
abs(δb) ≤ tol * abs(b) && break
end
println("Solution = ", b) # 3.14159268 / solution π= 3.14159265
# SOLUTION-2: use Optim to minimizing g(x)^2
using Optim
h(x) = ((g.(x)).^2)[1]
dh(x) = (2*g.(x).*f.(x))[1] # derivative
# Checks:
g(pi) ≈ 0 # true
h(pi) ≈ 0 # true
dh(pi) ≈ 0 # true
b = [1.0] # initial value / solution π= 3.14159265
result = optimize(h, b, Newton(), Optim.Options(g_tol = 1e-16))
Optim.minimizer(result) # 3.14159265
result = optimize(h, dh, b, Newton(); inplace=false) # using derivative
Optim.minimizer(result) # 3.1415795 # does not improve accuracy
# SOLUTION-3: use NLsolve to find roots
using NLsolve
dg(x) = f.(x) # derivative
# Check:
g([pi]) ≈ 0 # true
b = [1.0] # initial value / solution π= 3.14159265
nlsolve(g, b; ftol=1e-16) # 3.14159264
nlsolve(g, dg, b; ftol=1e-16) # 3.14159263 # same accuracy...
# SOLUTION-4: use DifferentialEquations and interpolation
using DifferentialEquations, Plots, Dierckx | {
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using DifferentialEquations, Plots, Dierckx
u0 = 0; tspan = (a, 4.) # initial conditions & timespan
du(u,p,t) = f(t) # define the system
prob = ODEProblem(du, u0, tspan) # define the problem
sol0 = solve(prob, dtmax=0.01) # solve it
i = argmin(abs.(sol0.u .- k)) # index of time-step the closest to solution
tsol = LinRange(sol0.t[i-1], sol0.t[i+1],20) # spline it around the solution
spl = Spline1D(tsol, sol0.(tsol) .- k)
b = mean(roots(spl)) # 3.1415926536
1 Like
Yes; it looks like the problem with your code is that Optim.optimize is expecting a vector of initial guesses, not a scalar. Try passing [b] instead of b.
However, this is a root-finding problem g(b) = 0. It is almost always a mistake to try to solve a root-finding problem by using an optimization algorithm to minimize |g|². You should use a root-finding algorithm like those in NLsolve.jl.
Moreover, in this case you know the derivative analytically, as I mentioned above, in which case you should certainly provide it to the root-finding algorithm.
However, for a 1d root-finding problem (b is a scalar) with an analytically known derivative, all of the generic root-finding packages will basically boil down to a Newton iteration like the one I wrote. The clever algorithms are mainly for the case where you don’t know the derivative, or have lots of derivatives (a big Jacobian) and can’t affort to compute them all or to invert the Jacobian.
Alternatively, in 1d, especially for smooth functions, there are often more clever algorithms available. For example, you could use ApproxFun.jl to construct a high-accuracy polynomial approximation of your integrand f, call cumsum to compute its integral, and use the ApproxFun.roots function to find all of the places where the integral equals k.
6 Likes | {
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6 Likes
Rather than minimize |g|², one could apply g(b)=0 as an equality constraint to an optimization algorithm that supports such constraints, e.g. NLopt.jl or JuMP.jl. I often find it convenient to deal with roots the same way I deal with optimization. Sometimes you can leave out an objective function altogether, or just specify a constant like 0, to be “minimized” subject to the constraints (to some tolerance). However, I’ve never compared performance to a root finder.
2 Likes
Tried to follow it the best possible and have updated code above for 3 different methods: Newton by yourself, Optim and NLsolve.
Including the derivative did not seem to help much, but any blame should surely be on me. BR
*PS: code above now runs but better not shaking it too much…
1 Like
Yes, solving the optimization problem \min_x 1 subject to the constraint g(x)=0 is, of course, exactly equivalent to the root-finding problem g(x)=0, and a sufficiently clever NLP algorithm will perform steps essentially equivalent to Newton steps (perhaps limited by a trust region). They almost certainly aren’t going to be better than Newton steps, however, and in many cases will be worse (because they may spend a lot of effort doing computational steps that are only needed for more general NLPs).
In a single-variable problem where the derivative is known, if you aren’t going to do something more clever (e.g. Chebyshev fits, complex analysis, continued fractions, etc.) you might as well just use Newton. (Either implementing it yourself — it’s just a few lines — or calling a canned implementation like the one in Roots.jl).
6 Likes
There is another way. Define R as
$$\int_a^{R(x)}f=x$$
you get
$$\frac{d}{dx}R(x) = \frac{1}{f(R(x))},\quad R(a)=0$$
You can use an ODE solver to get b=R(k). This is the main trick of my paper on PDMP.
@rveltz, it is only me, or your Latex is not displaying properly?
3 Likes
It gives:
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@rveltz, it is only me, or your Latex is not displaying properly?
3 Likes
It gives:
1 Like
@rveltz, mathematically this looks pretty much like @stevengj’s solution above. The difference is that the differential equations solver will not use Newton’s method?
Would like to try this when time allows, but would be surprised if it beats Newton in terms of accuracy and fast convergence.
I would say it is better in that you dont need to recompute \int_a^bf if you know that the solution b_{sol} is greater (smaller) than b. You “waste” less computation
1 Like
Yeah, it seems to me that there is a lot of redundancy in re-computing the full integral at each step. It should be enough to compute the integral only over the ‘correction interval’. (One should probably take some care with accumulated errors, however.)
2 Likes
@rveltz, in your nice solution how do you handle the cases with singularities where the function f() has zeros and 1/f() explodes to ∞?
Hum, I dont know! I guess the Newton solver will sufer as well. I’d say you have to use a stiff ODE solver that can handle finite time explosion.
1 Like
The topic “Solve equation” doesn’t do justice to the interesting things discussed here. @carloslesmes can I suggest editing the topic title to something more specific? For example, “Solve for upper/lower bound of a numerical definite integral” or some such.
1 Like
Am I looking at this crazy, or wouldn’t the correct boundary condition be R(0) = a instead of R(a) = 0?
1 Like
nope you are right! Thank you
Newton did not suffer for f(x)=exp(x)sin(x) in the example above with a zero at the solution pi. However, someone else is suffering to have your method working for that f(x) | {
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# Interpolation¶
In [2]:
import numpy as np # numerics
import matplotlib.pyplot as plt # plotting
%matplotlib inline
## Key concept¶
• Given two arrays of $x_g$, $y_g$ data points (e.g., a plot of points).
• You want to estimate the $x_w$, $y_w$ value of an intermediate data point.
• You know the value of $x_w$ (cause it's where you want the point).
• You need to interpolate to find the corresponding $y_w$.
In [3]:
#### Just plotting some data to visualize
x_given = np.array([0,1,2,3,4,5,6,7,8,9,10])
y_given = np.cos(x_given**2.0/8.0) + 1
plt.rc('font', size=16)
plt.plot(x_given, y_given, 'o:')
plt.plot([2.5,2.5],[-0.1,1.6], '--', color='gray')
plt.plot([-0.1,2.5],[1.6,1.6], '--', color='gray')
plt.plot([2.5], [1.6], '*', markersize=20)
plt.xlim([-0.1,11]); plt.ylim([-0.1,2.2])
plt.xlabel('x'); plt.ylabel('y')
plt.text(2.6,0, r"$x_w$", fontsize=16); plt.text(0,1.7, r"$y_w$", fontsize=16);
### Question¶
• How to get intermediate values, that is, values between those that are part of the given data?
• Fit a curve to the data, then evaluate from that curve.
• Take the closest data point
• Linear interpolation
• Higher order interpolation
• (Others?)
### Question¶
• How does curve fitting differ from interpolation?
• Normally, interpolation is done locally, between a few (often two) bounding points.
• Curve fitting involves making a best fit curve everywhere
• Usually, a best-fit curve will not pass through all data points.
• Useful when there is noise and we want the underlying curve, or a model equation for the data.
• Interpolation normally considers the points themselves, not a single curve through everything.
• Even with noisy data, we might want to interpolate an intermediate point.
• If you want an intermediate point to some given data, use interpolation.
• If you want to fit a model though some scattered data, and then evaluate the model, use curve fitting.
### Examples?¶ | {
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### Examples?¶
• Think of some examples where you might use this.
## Linear Interpolation Exercise¶
• Given the xg and yg data below.
• Write a function called Linterp that takes the following arguments:
• xg an array of given x data
• yg an array of given y data corresponding to xg
• xw the value of x we want to interpolate at
• The function returns yw corresponding to xw
• Assume xg are uniformly spaced, and ascending.
Questions
• What is the linear interpolation formula at a given location?
• We write the equation for a line based on two points, then evaluate the line at the intermediate point:
• equate slopes, then solve for y_w: $$\frac{y_w-y_0}{x_w-x_0} = \frac{y_1-y_0}{x_1-x_0},$$ $$y_w = y_0 + (x_w-x_0)\frac{y_1-y_0}{x_1-x_0}.$$
In [4]:
#-------------------- Set some "given" x, y data, (normally given to us)
xg = np.array([0,1,2,3,4,5,6,7,8,9,10.]) # given x data
yg = np.cos(xg**2.0/8.0)+1 # given y data
print("xg = "+np.array2string(xg, formatter={'float_kind':lambda x: f"{x:4.2f}"}))
print("yg = "+np.array2string(yg, formatter={'float_kind':lambda x: f"{x:4.2f}"}))
#-------------------- interpolate to xw=2.5
xw = 2.5
xg = [0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00]
yg = [2.00 1.99 1.88 1.43 0.58 0.00 0.79 1.99 0.85 0.24 2.00]
### New library¶
In [4]:
from scipy.interpolate import interp1d
• interp1d takes the given x array and the given y array as arguments.
• Returns a function.
• Call that function wherever you want to interpolate to.
In [14]:
xg = np.array([0,1,2,3,4,5,6,7,8,9,10]) # given x data
yg = np.cos(xg**2.0/8.0)+1 # given y data
#---------------
f_interp = interp1d(xg, yg)
#---------------
xw = 2.5
yw = f_interp(xw)
print(yw)
1.6543795393445195
### Exercise¶ | {
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#---------------
xw = 2.5
yw = f_interp(xw)
print(yw)
1.6543795393445195
### Exercise¶
• Take the previous xg, yg data and plot the data as points.
• Plot 1000 points of the underlying function used to get xg and yg, called xx
• Also, plot values of a linear interpolant at the same 1000 points.
In [5]:
xg = np.array([0,1,2,3,4,5,6,7,8,9,10]) # given x data
yg = np.cos(xg**2.0/8.0)+1 # given y data
#### Try replacing¶
f_interp = interp1d(xg, yg)
#### with this¶
f_interp = interp1d(xg, yg, kind='cubic')
• This uses a cubic spline interpolant instead of a linear interpolant.
• That is, we use cubic functions between points that match up smoothly at interfaces.
Try this:
• Interpolate an x value that is outside of the bounds of the given xg data.
• What happens?
### Extrapolation¶
• As listed above, you will get an error if you try to call f_interp with an x value that is outside of the upper and lower bounds of the original xg array.
• This can be avoided using the fill_value='extrapolate' argument, like so:
• f_interp = interp1d(xg, yg, fill_value='extrapolate'
## Summary¶
1. Import library
2. You have some data from somewhere, xg, yg
3. Get ther interpolant function fi=interp1d(xg, yg)
4. Set desired xw intermediate points to interpolate at (can be an array).
5. Perform the interpolation to yw=fi(xw)
In [6]:
from scipy.interpolate import interp1d
xg = np.array([1,2,3,4,5]) # some data you have
yg = np.array([11, 2.2, 3.3, -88, 9])
fi = interp1d(xg,yg)
xw = 2.5
yw = fi(xw)
## Question¶
• What if you have given xg and yg arrays and you know the yw and want to interpolate to the corresponding xw?
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# Big Oh and Big Omega when $n$ and $\log n$ terms are in $f(n)$
having problems with big oh and big omega functions when there is a $$\log n$$ added or subtracted. For example how do I deal with $$n+\log n$$ or $$n-\log n$$ when I have to determine whether the function is in $$\Omega(n)$$ or in $$\Omega(n^2)$$? For example, is $$n-\log n$$ in $$\Omega(n)$$ or in $$\Omega(n^2)$$?
I cannot ignore the log function and am not sure how to deal with it. Polynomials and logs when multiplied I find OK. But I have a mental block over this one so help would be appreciated
You are right that you cannot a priori ignore additional terms, although morally you can as the "smaller" terms do not contribute to the asymptotic growth.
As an example $$f(n) :=n+\log n$$ is in $$\Theta(n)$$. Why? Going back to the definition, we want to show that $$f(n)$$ is in $$O(n)$$ and in $$\Omega(n)$$. Showing $$f(n)\in \Omega(n)$$ is immediate, as $$f(n) \geq n$$ for all $$n>0$$.
To show that $$f(n)$$ is in $$O(n)$$, simply notice that for large enough $$n$$ (say $$n>N$$ for some constant $$N$$) we have $$\log n \leq n$$ and thus $$f(n)\leq 2n$$ for all $$n>N$$. By definition $$f(n)\in O(n)$$.
Actually, you CAN ignore the log function in a sum or substraction, because the log is always asymptotically negligible in front of $$n^x$$, for every $$x > 0$$.
If you consider the function $$f(n) = n^x + \alpha\log n$$ (where $$x > 0$$ and $$\alpha \in \mathbb{R}$$), then there exists $$A, B \in \mathbb{R}_+^*$$ such that $$An^x \leq f(n) \leq Bn^x$$. That means that $$f\in \Omega(n^x)$$ and $$f\in \mathcal{O}(n^x)$$. | {
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• Agreed with the f(n) =n^2 or cubed -logn as the polynomial grows faster we can ignore the log n. f(n) grows faster than log n. Is it sufficient then to just prove that f(n) =n and is in Big omega and ignore the log term on the basis that as Tassle says logn<n . is this sufficient for a formal proof Feb 22 '21 at 15:40
• Sorry, I didn't really get your question… I'll try to answer what I understood. When you have a sum $f(n) = g(n) + h(n)$ with $h \in o(g)$ (see en.wikipedia.org/wiki/Big_O_notation#Little-o_notation), then $f\in \Omega(g)$ and $f\in \mathcal{O}(g)$. This is the case in your question, since $\log \in o(n^x)$ for $x > 0$, and yes it is sufficient for a proof. Feb 22 '21 at 15:48
In your example, $$n-\log(n)=\theta(n)$$, that is $$n-\log(n) = \Omega(n)$$ and also $$n-\log(n)=O(n)$$.
You can see this like that: | {
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You can see this like that:
• $$n-\log(n)\le n = O(n)$$
• $$n-\log(n) \ge n - 0.5n = \Omega(n)$$
• as n-logn is always less than n the function can only be in Big Oh . It cannot be in Big omega as if I divide by n^2 to fiind c I get 1-logn > or equal to c. As n gets greater than 2 n-logn gets more negative and c is no longer a constant so it cannot be in Big Omega. Am I on the right track? Feb 24 '21 at 9:09
• From a certain point $log(n)\le0.5n$. This means that $n-\log(n)$ will actually not be negative, but actually if you substitute the inequality you get $n-\log(n)\ge 0.5n$, which is $\Omega(n)$. However, you are correct that it is not $\Omega(n^2)$. Feb 24 '21 at 10:20
• sorry. I meant n-logn is not in Big Omega (n) not Big Omega (n^2). My original question is to prove (n-logn) = Ω(n) true or false. I was trying to prove it false by contradiction. Hence my reasoning of c needing to be a positive constant for all n> no.I am not sure where you came up with the 0.5n either. I plotted the graphs and n-logn is always under f(n)= n when n>1 Feb 24 '21 at 10:41
• As my answer showed, it is indeed always lower than $n$. But, in the same time, it is bigger than $0.5n$. Try to draw this graph as well Feb 24 '21 at 11:32
• You are right and thank you for your help . In fact from what you have said I can prove that n-logn =big theta(n) . Feb 24 '21 at 12:07 | {
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# Help understanding why a complete, totally bounded metric space implies every infinite subset has a limit point
I'm reading the following proof. Properties II and III are in my title, that a complete, totally bounded metric space implies every infinite subset has a limit point.
I have two questions near the end. Why does $d(x_n,x)\lt 2/n$? And secondly, why is $x$ a limit point of $A$? What other point in the neighborhood of $x$ is also in $A$? I don't get why they mention that $3/n\to 0$ as $n\to\infty$. How does that imply it's a limit point?
-
Please give a reference when you quote text from somewhere else. Otherwise, how can anyone else look it up? – Carl Mummert Sep 25 '11 at 1:53 | {
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There’s a slight error in the proof: the claim should be that $d(x_n,x) \le 2/n$. To see this, let $\epsilon$ be any positive real number; the sequence of $x_m$’s converges to $x$, so there is a positive integer $m > n$ such that $d(x_m,x) < \epsilon$. By the triangle inequality $$d(x_n,x) \le d(x_n,x_m)+d(x_m,x) < \frac2n + \epsilon.\tag{1}$$ Thus, $$d(x_n,x) < \frac2n + \epsilon$$ for every $\epsilon > 0$, and hence $d(x_n,x) \le \dfrac2n$.
This small error doesn’t affect the next step of the argument: if $y \in B(x_n,1/n)$, then $d(x_n,y) < 1/n$, so $$d(x,y) \le d(x,x_n)+d(x_n,y) \le \frac2n + d(x_n,y) < \frac2n + \frac1n = \frac3n,$$ $y \in B(x,3/n)$, and therefore $B(x_n,1/n) \subseteq B(x,3/n)$.
Now $B(x_n,1/n)\cap A$ is infinite for each $n$, and $B(x_n,1/n) \subseteq B(x,3/n)$, so $B(x,3/n)\cap A$ is infinite for each $n$. Since $3/n\to 0$ as $n\to\infty$, for any $\epsilon > 0$ there is an $n_\epsilon$ such that $3/n_\epsilon < \epsilon$. But then $B(x,\epsilon)\cap A \supseteq B(x,3/n_\epsilon)\cap A$, which is infinite. Thus, every nbhd of $x$ contains infinitely many points of $A$.
Added: To clarify, it is in fact true that $d(x_n,x)<2/n$ for each $n$; it just doesn’t follow directly from the fact that $d(x_m,x_n) < 2/m$ when $m<n$, as the weaker inequality does. If we want the strict inequality, we can modify the argument that I gave by choosing $m>n$ so that $1/m<\epsilon/2$ and $d(x_m,x)<\epsilon/2$ and then replacing $(1)$ by $$d(x_n,x)\le d(x_n,x_m)+d(x_m,x)<\frac1n+\frac1m+\frac{\epsilon}{2}<\frac1n+\epsilon.$$ Since $\epsilon$ can be chosen arbitrarily small, we conclude that $d(x_n,x)\le 1/n$ for each $n$. | {
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I don't think this is an error. They had established $d(x_n,x_m) \leq 1/m+1/n \lt 2/n$ for $n \lt m$ already (with $n$ and $m$ interchanged), so if you fix $n$ and let $m \to \infty$ then you get strict inequality. – commenter Sep 25 '11 at 1:01 @commenter: Their statement isn’t false, but it doesn’t follow from the fact that $d(x_n,x_m)<2/m$ when $m I think it worth settling some terminology. This is from Topology: a first course by James R. Munkres. He identifies three versions of compactness for a topological space: (A) compactness, then (B) your property (which he calls "limit point compactness") and a milder condition (C) he calls "sequential compactness" which is that every infinite sequence of points has a convergent subsequence. He proves that (A) implies (B) implies (C). He also proves that the three conditions are equivalent for a metric space. Finally, a well-known theorem is that a metric space is compact if and only if it is complete and totally bounded. So the strongest condition is what is usually discussed in this setting. However, (A) then implies (B). Finally, and this is not obvious, the product of two compact topological spaces is compact, and the product of two sequentially compact spaces is sequentially compact. What is unexpected is that the product of two limit point compact spaces need not be limit point compact. However, if they are metric spaces, the result does hold then. - To address the first question (the other is addressed in the post by Brian). [Also, there's no error.] Consider$\epsilon=1/(2n)$. Then there is some$N>0$such that$d(x_m,x)<1/(2n)$for all$m \geq N$. Let$m=\max\{N,2n\}$. Then$d(x_n,x) \leq d(x_n,x_m)+d(x_m,x) < [(1/n)+(1/m)]+1/(2n) \leq (1/n) + (1/2n) + (1/2n) = 2/n$. - While the assertion in question is true, the presentation is in my view seriously flawed, in that it suggests a justification that doesn’t in fact work. – Brian M. Scott Sep 25 '11 at 1:36 I'm not sure this is quite what you're looking for, but I believe | {
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Brian M. Scott Sep 25 '11 at 1:36 I'm not sure this is quite what you're looking for, but I believe the author just jumped from$d(x_n,x_m) < 2/n$for$n | {
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Hint: Find the equation of the line passing through the two given points. This calculator can find the center and radius of a circle given its equation in standard or general form. What is the Equation of a Circle? Different forms equations of straight lines. Equation of circle when endpoints of the diameter are given : (x - x 1) (x - x 2) + (y - y 1) (y - y 2) = 0. Enter Circle Equation Find the equation of a circle that has a diameter with the endpoints given by the points A(1,1) and B(2,4) Step 1: Find the Midpoint ( h , k ) of AB : Write Standard Form When Given Two Points You. How To Calculate The Equation Of A Circle From 3 Points Tessshlo. Also, it can find equation of a circle given its center and radius. Given circle A with the equation That circle has center (-2,0) and radius √5 a)show that the circle passes through point (0,1) That is true so it proves that it passes through that point. Ex Find The Point On A Circle Given An Angle And Radius You. Two point form This online calculator can find and plot the equation of a straight line passing through the two points. Important Properties: † Equation of a circle: An equation of the circle with center (h;k) and radius r is given … Here we are going to see how to find the equation of circle with extremities of diameter are given. I did that and found the radius was 5 but where do I go from there? $\Rightarrow {g^2} + {f^2} – c = \frac{{{{\left( { – ag – bf + d} \right)}^2}}}{{{a^2} + {b^2}}}\,\,\,{\text{ – – – }}\left( {\text{v}} \right)$ Given two points, A(-2,5) and B(-4,3) find the equation of the circle with diameter AB.? I really need help with this question :(, Im sure you need to find the diameter so you can find the radius. † Circle: is the set of all points in a plane that lie a flxed distance from a flxed point. It is straightforward BUT a bit trickier than you expect! Circle Calc Find C D A R. Equation Of A Circle Through Three Points You. Point A (9, 2) Point B (3, -4) Point C (5, -6) Let's put these | {
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# How to find a standard deviation determined by a Normal distribution probability?
The question is
A liquid drug is marketed in phials containing a nominal 1.5ml but the amounts can vary slightly. The volume in each phial may be modeled by a normal distribution with the mean 1.55ml and standard deviation $$\sigma$$ ml. The phials are sold in packs of 5 randomly chosen phials . It is required that in less than 0.5% of the packs will the total volume of the drug be less than 7.5ml. Find the greatest possible value of $$\sigma$$.
I need to find the greatest possible value of the standard deviation ($$\sigma$$). I worked out the following:
$$\mu= 1.55*5 = 7.75.$$
We are asked to find value of $$\sigma$$ such that probability of (total volume of $$5$$ packs $$\lt 7.5)\lt0.5\%$$
$$P(X\lt7.5)\lt0.005.$$
After standardizing, $$P(X\le\frac{7.5-7.75}{\sigma/5})<0.005$$ and I found $$\sigma=0.2170.$$ However, the answer provided is $$0.0434.$$
• Please add the self-study tag, read its tag-wiki, and indicate the specific help you need at the point you struck difficulty. Jun 9 '16 at 2:37
• what's CTL? ... ... Also please check the details of the question, it looks like you may have a mistake somewhere. Where did the 7.75 in your working come from? Please show more detail/explanation of what you're doing. (As far as possible your responses should result in edits to your question) Jun 9 '16 at 2:38
• How have you approached/engaged it so far? Any partly successful paths? Where else have you looked for answers? Jun 14 '16 at 2:06
• Interestingly, neither answer is correct.
– whuber
Jun 14 '16 at 13:59
Among the objectives of good introductory statistics courses is learning how to think about the Normal distribution. This question provides a nice example.
The key is to use units of measurement that are adapted to the distribution. That is, let the mean be the zero point and let the standard deviation be one unit. This is what a "Z score" measures. | {
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In light of this, let's parse the question. To do so, I will use two fundamental facts: expectations add ("linearity of expectation") and variances of independent variables also add:
• The mean volume of one pack is 1.55 ml, whence the mean volume of five packs must be five times as large, or 7.75 ml: this is the zero point.
• Since the unknown variance of a single pack is $$\sigma^2,$$ the variance of the sum of five independent packs is $$5\sigma^2.$$ Therefore the standard deviation of the sum--the unit of measurement we must adopt--is $$\sqrt{5\sigma^2} = \sigma\sqrt{5}.$$
The question stipulates that in less than 0.5% of cases should the total be less than 7.5 ml. For the (standard) Normal distribution we remember (or can compute) that exactly 0.5% of cases are $$2.57\ldots$$ or more less than the mean. An example of this computation is
qnorm(0.5/100)
in R or
=NORMSINV(0.5/100)
in Excel, for instance.
One aim of the introductory course is to help you reach the point where such considerations are automatic: you can do them in your head correctly, apart (perhaps) from the arithmetical calculations.
This preliminary work enables us to rephrase the question like this:
What unit of measurement, given by $$\sigma\sqrt{5}$$ for a five-pack of drugs, will re-express an amount of $$7.5$$ ml as being $$2.57$$ less than $$7.75$$ ml?
The solution obviously is
$$\sigma\sqrt{5} = (7.75 - 7.5)/2.57\ldots = 0.097\ldots,$$
implying
$$\sigma = \frac{0.097\ldots}{\sqrt{5}} = 0.0433797\ldots$$
Comparing this result to the question shows that the work in the question was entirely correct up to the point where "$$\sigma/5$$" appeared: the square root was lost. This suggests remembering to think in terms of variances rather than standard deviations. | {
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Comparing this result to the older answers that were posted also shows how they were basically moving in the correct direction but made mistakes along the way, too. Because arithmetical mistakes are easy to make, when one has the chance it's a good idea to check probabilistic calculations with simulations. For instance, the following R statement generates a large number of five-packs of drugs as described in the question (using the answer I obtained) and, to check my answer, computes the fraction with totals less than 7.5 ml:
mean(colSums(matrix(rnorm(5*1e6, 1.55, -0.25/qnorm(0.5/100) / sqrt(5)), nrow=5)) < 7.5)
(You can see all the data from the question embedded in this expression, along with the value 1e6 giving the number of five-packs to simulate.) When I run and re-run this code (which takes less than a second each time), I consistently obtain results between 0.0048 (0.48%) and 0.0052 (0.52%), in satisfactory agreement with the intended 0.5% target.
I think your understanding of the variance of the sum is mistaken. The variance of the 5-pack sum is 25 times the variance of the single pack.
• The only way you could justify the factor of $25$ is to suppose the five packs are perfectly correlated. Assuming, as is more likely the intent, that the five-pack sum can be modeled as the total of five independent Normal variables $X_1+\cdots+X_5$, its variance will be $$\operatorname{Var}(X_1+\cdots+X_5)=\sigma^2 +\cdots+\sigma^2=5\sigma^2.$$ Consequently its standard deviation will be $\sigma\sqrt{5}$, which (by following the path outlined in the question) leads directly to the correct answer.
– whuber
Jun 14 '16 at 14:01
\begin{align} 5\times 1.55 &= 7.75 \\ 5\times SD &= 5SD \end{align} Problem statement: $$P(X<7.5)<0.005$$ \begin{align} \frac{(7.5-7.75)}{(5SD^2)^{1/2}} &< 0.005 \\[8pt] \frac{-0.25}{2.576} &= (5SD^2)^{1/2} \\[5pt] &-0.0970 / 5^{(1/2)} \end{align} $$0.0434$$ as the standard deviation can never me negative, take the mod value. | {
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• Welcome to Stats.SE. You may give hints but please do not give the full answer. Furthermore, can you please edit your post and explain the key steps in the solution and use MathJax in the formulas? Apr 26 '19 at 10:51 | {
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# Math Help - Problem solving - sucessive tetrahedral numbers
1. ## Problem solving - sucessive tetrahedral numbers
would you guys be able to give me a hand with this problem solving question, i have no idea where to start...
Thanks
Zac
2. Hello, Zac!
12. The first four tetrahedral numbers are: . $1,\;4,\;10,\;20$
Find the pattern, then predict the next 3 terms of the sequence.
Take the difference of consecutive pairs of terms.
$\begin{array}{cccccccc}\text{Sequence:} & 1 && 4 && 10 && 20 \\
\text{Difference:} & & 3 & & 6 & & 10 \end{array}$
They are adding on consecutive "triangular" numbers.
. . $\begin{array}{ccc}3 & = & 1+2 \\ 6 &=&1+2+3\\ 10 &=&1+2+3+4 \end{array}$
We will add on the next three triangular numbers:
. . $\begin{array}{ccc}1+2+3+4+5 &=& 15 \\ 1+2+3+4+5+6 &=& 21 \\ 1+2+3+4+5+6+7 &=& 28\end{array}$
$\begin{array}{cccccccccccccc}\text{Sequence:} & 1 && 4 && 10 && 20 && {\color{blue}35} && {\color{blue}56} && {\color{blue}84}\\
\text{Difference:} & & 3 & & 6 & & 10 && +15 && +21 && +28\end{array}$
What is the $n^{th}$ term of the sequence?
I'll skip all the algebra . . .
$\text{The }n^{th}\text{ term is: }\;a_n \;=\;\frac{n(n+1)(n+2)}{6}$
3. Originally Posted by Soroban
I'll skip all the algebra . . .
$\text{The }n^{th}\text{ term is: }\;a_n \;=\;\frac{n(n+1)(n+2)}{6}$
as this is a problwm solving questions and according to my sheet i must show the algebra would you mind putting it up?
thanks so much!
Zac
4. Hello, Zac!
This will take a while . . . better sit down.
Take differences of consecutive terms,
. . then take differences of the differences, and so on.
$\begin{array}{cccccccccccccc}\text{Sequence} & 1 && 4 && 10 && 20 && 35 && 56 && 84 \\
\text{1st diff.} & & 3 && 6 && 10 && 15 && 21 && 28 \\
\text{2nd diff.} & & & 3 && 4 && 5 && 6 && 7 \\
\text{3rd diff.} & & & & 1 && 1 && 1 && 1 \end{array}$
The third differences are constant.
. . This tells us that the generating fuction is a cubic. | {
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The third differences are constant.
. . This tells us that the generating fuction is a cubic.
The general cubic function is: . $f(n) \;=\;an^3 + bn^2 + cn + d$
We will use the first four terms of the sequence . . .
$\begin{array}{ccccc}f(1)\:=\:1\!: & a + b + c + d & = & 1 & [1] \\
f(2) \:=\:4\!: & 8a + 4b + 2c + d &=& 4 & [2] \\
f(3) \:=\:10\!: & 27a + 9b + 3c + d &=& 10 & [3] \\
f(4) \:=\:20\!: & 64a + 16b + 4c + d &=& 20 & [4] \end{array}$
$\begin{array}{ccccc}\text{Subtract [2] - [1]:} & 7a + 3b + c &=& 3 & [5] \\
\text{Subtract [3] - [2]:} & 19a + 5b + c &=& 6 & [6] \\
\text{Subtract [4] - [3]:} & 37a + 7b + c &=& 10 & [7] \end{array}$
$\begin{array}{ccccc}\text{Subtract [6] - [5]:} & 12a + 2b &=& 3 & [8] \\
\text{Subtract [7] - [6]:} & 18a + 2b &=& 4 & [9] \end{array}$
$\begin{array}{ccccccc}\text{Subtract [9] - [8]:}& 6a \;=\; 1 & \Rightarrow & \boxed{a \;=\;\frac{1}{6}} \end{array}$
Substitute into [8]: . $12\left(\frac{1}{6}\right) + 2b \:=\:3\quad\Rightarrow\quad\boxed{ b \:=\:\frac{1}{2}}$
Substitute into [5]: . $7\left(\frac{1}{6}\right) + 3\left(\frac{1}{2}\right) + c \:=\:3 \quad\Rightarrow\quad \boxed{c\:=\:\frac{1}{3}}$
Substitute into [1]: . $\frac{1}{6} + \frac{1}{2} + \frac{1}{3} + d \:=\:1 \quad\Rightarrow\quad\boxed{ d\:=\:0}$
Hence, the function is: . $f(n) \;=\;\frac{1}{6}n^3 + \frac{1}{2}n^2 + \frac{1}{3}n \;=\;\frac{n}{6}(n^2+3n+ 2)$
. . Therefore: . $\boxed{f(n) \;=\;\frac{n(n+1)(n+2)}{6}}$ | {
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# Value of $\sum\limits_n x^n$
Why is $\displaystyle \sum\limits_{n=0}^{\infty} 0.7^n$ equal $1/(1-0.7) = 10/3$ ?
Can we generalize the above to
$\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ?
Are there some values of $x$ for which the above formula is invalid?
What about if we take only a finite number of terms? Is there a simpler formula?
$\displaystyle \sum_{n=0}^{N} x^n$
Is there a name for such a sequence?
This is being repurposed in an effort to cut down on duplicates, see here: Coping with abstract duplicate questions.
and here: List of abstract duplicates.
-
This is a geometric series. See en.wikipedia.org/wiki/Geometric_series#Formula – lhf Mar 25 '11 at 15:47
Please replace the starting index $n=1$ by $n=0$ (otherwise the resulting sum is $0.7$ times what you wrote). – Did Mar 25 '11 at 17:59
@Jonas: Fair point: done. – Arturo Magidin Mar 25 '11 at 19:18
@Didier: That was my fault upon edit; fixed. – Arturo Magidin Mar 25 '11 at 19:19
@Arturo: Thanks. Sorry, I shouldn't have deleted my comment; I did so after I saw you had changed the title, and only then saw your comment. (To anyone who is confused: I had pointed out that including "geometric series" in the title gave additional context that made lhf's and yoyo's first sentences seem a little strange.) – Jonas Meyer Mar 25 '11 at 19:41
By definition, a "series" (an "infinite sum") $$\sum_{n=k}^{\infty} a_n$$ is defined to be a limit, namely $$\sum_{n=k}^{\infty} a_n= \lim_{N\to\infty} \sum_{n=k}^N a_n.$$ That is, the "infinite sum" is the limit of the "partial sums", if this limit exists. If the limit exists, equal to some number $S$, we say the series "converges" to the limit, and we write $$\sum_{n=k}^{\infty} a_n = S.$$ If the limit does not exist, we say the series diverges and is not equal to any number.
So writing that $$\sum_{n=0}^{\infty} 0.7^n = \frac{1}{1-0.7}$$ means that we are asserting that $$\lim_{N\to\infty} \sum_{n=0}^N0.7^n = \frac{1}{1-0.7}.$$ | {
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So what your question is really asking is: why is this limit equal to $\frac{1}{1-0.7}$? (Or rather, that is the only way to make sense of the question).
In order to figure out the limit, it is useful (but not strictly necessary) to have a formula for the partial sums, $$s_N = \sum_{n=0}^N 0.7^n.$$ This is where the formulas others have given come in. If you take the $N$th partial sum and multiply by $0.7$, you get $$\begin{array}{rcrcrcrcrcrcl} s_N &= 1 &+& (0.7) &+& (0.7)^2 &+& \cdots &+& (0.7)^N\\ (0.7)s_N &= &&(0.7) &+& (0.7)^2 &+&\cdots &+&(0.7)^N &+& (0.7)^{N+1} \end{array}$$ so that $$(1-0.7)s_N = s_N - (0.7)s_N = 1 - (0.7)^{N+1}.$$ Solving for $s_N$ gives $$s_N = \frac{1 - (0.7)^{N+1}}{1-0.7}.$$ What is the limit as $N\to\infty$? The only part of the expression that depends on $N$ is $(0.7)^{N+1}$. Since $|0.7|\lt 1$, then $\lim\limits_{N\to\infty}(0.7)^{N+1} = 0$. So, $$\lim_{N\to\infty}s_N = \lim_{N\to\infty}\left(\frac{1-(0.7)^{N+1}}{1-0.7}\right) = \frac{\lim\limits_{N\to\infty}1 - \lim\limits_{N\to\infty}(0.7)^{N+1}}{\lim\limits_{N\to\infty}1 - 0.7} = \frac{1 - 0}{1-0.7} = \frac{1}{1-0.7}.$$ Since the limit exists, then we write $$\sum_{n=0}^{\infty}(0.7)^n = \frac{1}{1-0.7}.$$
More generally, a sum of the form $$a + ar + ar^2 + ar^3 + \cdots + ar^k$$ with $a$ and $r$ constant is said to be a "geometric series" with initial term $a$ and common ratio $r$. If $a=0$, then the sum is equal to $0$. If $r=1$, then the sum is equal to $(k+1)a$. If $r\neq 1$, then we can proceed as above. Letting $$S = a +ar + \cdots + ar^k$$ we have that $$S - rS = (a+ar+\cdots+ar^k) - (ar+ar^2+\cdots+a^{k+1}) = a - ar^{k+1}$$ so that $$(1-r)S = a(1 - r^{k+1}).$$ Dividing through by $1-r$ (which is not zero since $r\neq 1$), we get $$S = \frac{a(1-r^{k+1})}{1-r}.$$ | {
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A series of the form $$\sum_{n=0}^{\infty}ar^{n}$$ with $a$ and $r$ constants is called an infinite geometric series. If $r=1$, then $$\lim_{N\to\infty}\sum_{n=0}^{N}ar^{n} = \lim_{N\to\infty}\sum_{n=0}^{N}a = \lim_{N\to\infty}(N+1)a = \infty,$$ so the series diverges. If $r\neq 1$, then using the formula above we have: $$\sum_{n=0}^{\infty}ar^n = \lim_{N\to\infty}\sum_{n=0}^{N}ar^{N} = \lim_{N\to\infty}\frac{a(1-r^{N+1})}{1-r}.$$ The limit exists if and only if $\lim\limits_{N\to\infty}r^{N+1}$ exists. Since $$\lim_{N\to\infty}r^{N+1} = \left\{\begin{array}{ll} 0 &\mbox{if |r|\lt 1;}\\ 1 & \mbox{if r=1;}\\ \text{does not exist} &\mbox{if r=-1 or |r|\gt 1} \end{array}\right.$$ it follows that: \begin{align*} \sum_{n=0}^{\infty}ar^{n} &=\left\{\begin{array}{ll} 0 &\mbox{if a=0;}\\ \text{diverges}&\mbox{if a\neq 0 and r=1;}\\ \lim\limits_{N\to\infty}\frac{a(1-r^{N+1})}{1-r} &\mbox{if r\neq 1;}\end{array}\right.\\ &= \left\{\begin{array}{ll} \text{diverges}&\mbox{if a\neq 0 and r=1;}\\ \text{diverges}&\mbox{if a\neq 0, and r=-1 or |r|\gt 1;}\\ \frac{a(1-0)}{1-r}&\mbox{if |r|\lt 1;} \end{array}\right.\\ &=\left\{\begin{array}{ll} \text{diverges}&\mbox{if a\neq 0 and |r|\geq 1;}\\ \frac{a}{1-r}&\mbox{if |r|\lt 1.} \end{array}\right. \end{align*}
Your particular example has $a=1$ and $r=0.7$.
Since this recently came up (09/29/2011), let's provide a formal proof that $$\lim_{N\to\infty}r^{N+1} = \left\{\begin{array}{ll} 0 &\mbox{if |r|\lt 1;}\\ 1 & \mbox{if r=1;}\\ \text{does not exist} &\mbox{if r=-1 or |r|\gt 1} \end{array}\right.$$ | {
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If $r\gt 1$, then write $r=1+k$, with $k\gt0$. By the binomial theorem, $r^n = (1+k)^n \gt 1+nk$, so it suffices to show that for every real number $M$ there exists $n\in\mathbb{N}$ such that $nk\gt M$. This is equivalent to asking for a natural number $n$ such that $n\gt \frac{M}{k}$, and this holds by the Archimedean property; hence if $r\gt 1$, then $\lim\limits_{n\to\infty}r^n$ does not exist. From this it follows that if $r\lt -1$ then the limit also does not exist: given any $M$, there exists $n$ such that $r^{2n}\gt M$ and $r^{2n+1}\lt M$, so $\lim\limits_{n\to\infty}r^n$ does not exist if $r\lt -1$.
If $r=-1$, then for every real number $L$ either $|L-1|\gt \frac{1}{2}$ or $|L+1|\gt \frac{1}{2}$. Thus, for every $L$ and for every $M$ there exists $n\gt M$ such that $|L-r^n|\gt \frac{1}{2}$ proving the limit cannot equal $L$; thus, the limit does not exist. If $r=1$, then $r^n=1$ for all $n$, so for every $\epsilon\gt 0$ we can take $N=1$, and for all $n\geq N$ we have $|r^n-1|\lt\epsilon$, hence $\lim\limits_{N\to\infty}1^n = 1$. Similarly, if $r=0$, then $\lim\limits_{n\to\infty}r^n = 0$ by taking $N=1$ for any $\epsilon\gt 0$.
Next, assume that $0\lt r\lt 1$. Then the sequence $\{r^n\}_{n=1}^{\infty}$ is strictly decreasing and bounded below by $0$: we have $0\lt r \lt 1$, so multiplying by $r\gt 0$ we get $0\lt r^2 \lt r$. Assuming $0\lt r^{k+1}\lt r^k$, multiplying through by $r$ we get $0\lt r^{k+2}\lt r^{k+1}$, so by induction we have that $0\lt r^{n+1}\lt r^n$ for every $n$.
Since the sequence is bounded below, let $\rho\geq 0$ be the infimum of $\{r^n\}_{n=1}^{\infty}$. Then $\lim\limits_{n\to\infty}r^n =\rho$: indeed, let $\epsilon\gt 0$. By the definition of infimum, there exists $N$ such that $\rho\leq r^N\lt \rho+\epsilon$; hence for all $n\geq N$, $$|\rho-r^n| = r^n-\rho \leq r^N-\rho \lt\epsilon.$$ Hence $\lim\limits_{n\to\infty}r^n = \rho$. | {
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In particular, $\lim\limits_{n\to\infty}r^{2n} = \rho$, since $\{r^{2n}\}_{n=1}^{\infty}$ is a subsequence of the converging sequence $\{r^n\}_{n=1}^{\infty}$. On the other hand, I claim that $\lim\limits_{n\to\infty}r^{2n} = \rho^2$: indeed, let $\epsilon\gt 0$. Then there exists $N$ such that for all $n\geq N$, $r^n - \rho\lt\epsilon$. Moreover, we can assume that $\epsilon$ is small enough so that $\rho+\epsilon\lt 1$. Then $$|r^{2n}-\rho^2| = |r^n-\rho||r^n+\rho| = (r^n-\rho)(r^n+\rho)\lt (r^n-\rho)(\rho+\epsilon) \lt r^n-\rho\lt\epsilon.$$ Thus, $\lim\limits_{n\to\infty}r^{2n} = \rho^2$. Since a sequence can have only one limit, and the sequence of $r^{2n}$ converges to both $\rho$ and $\rho^2$, then $\rho=\rho^2$. Hence $\rho=0$ or $\rho=1$. But $\rho=\mathrm{inf}\{r^n\mid n\in\mathbb{N}\} \leq r \lt 1$. Hence $\rho=0$.
Thus, if $0\lt r\lt 1$, then $\lim\limits_{n\to\infty}r^n = 0$.
Finally, if $-1\lt r\lt 0$, then $0\lt |r|\lt 1$. Let $\epsilon\gt 0$. Then there exists $N$ such that for all $n\geq N$ we have $|r^n| = ||r|^n|\lt\epsilon$, since $\lim\limits_{n\to\infty}|r|^n = 0$. Thus, for all $\epsilon\gt 0$ there exists $N$ such that for all $n\geq N$, $| r^n-0|\lt\epsilon$. This proves that $\lim\limits_{n\to\infty}r^n = 0$, as desired.
In summary, $$\lim_{N\to\infty}r^{N+1} = \left\{\begin{array}{ll} 0 &\mbox{if |r|\lt 1;}\\ 1 & \mbox{if r=1;}\\ \text{does not exist} &\mbox{if r=-1 or |r|\gt 1} \end{array}\right.$$ | {
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The argument suggested by Srivatsan Narayanan in the comments to deal with the case $0\lt|r|\lt 1$ is less clumsy than mine above: there exists $a\gt 0$ such that $|r|=\frac{1}{1+a}$. Then we can use the binomial theorem as above to get that $$|r^n| = |r|^n = \frac{1}{(1+a)^n} \leq \frac{1}{1+na} \lt \frac{1}{na}.$$ By the Archimedean Property, for every $\epsilon\gt 0$ there exists $N\in\mathbb{N}$ such that $Na\gt \frac{1}{\epsilon}$, and hence for all $n\geq N$, $\frac{1}{na}\leq \frac{1}{Na} \lt\epsilon$. This proves that $\lim\limits_{n\to\infty}|r|^n = 0$ when $0\lt|r|\lt 1$, without having to invoke the infimum property explicitly.
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I think you mean "which is not zero since $r\neq 1$." – robjb Mar 26 '11 at 23:20
@Rob: Yes; thank you. – Arturo Magidin Mar 27 '11 at 0:22
+1 As its rare to see math researchers caring for even elementary questions. – Please Delete Account Mar 27 '11 at 2:18
Ignore my edit, I seem to have potatoes on my eyes. Sorry. – Lagerbaer Mar 31 '11 at 5:05
Arturo, I will mention one more way of proving that $r^n$ converges to $0$ as $n \to \infty$ if $|r| < 1$ that looks somewhat similar to your proof in the divergent case. Take $|r| = \frac{1}{1+a}$ for some $a > 0$. Then $$|r^n - 0| = |r|^n = \frac{1}{(1+a)^n} \leq \frac{1}{1+an} \leq \frac{1}{an} ,$$ which can be made sufficiently small by taking $n$ large enough. – Srivatsan Sep 30 '11 at 4:13
it's called a geometric series. let $-1<x<1$ and let $S_n=1+x+x^2+...+x^n$. then $xS_n=x+...+x^{n+1}=S^n-1+x^{n+1}$. move stuff around to get $$S_n=\frac{1-x^{n+1}}{1-x}$$ take the limit as $n\to\infty$ (noting that $x^n\to0$ if $|x|<1$)
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In yoyo's answer, you should also take into account that $x^{n+1}\to 0$ as $n\to\infty$ if $|x| < 1$. – knightofmathematics Mar 25 '11 at 16:48
...and $0^0=1$ (: – yoyo Mar 25 '11 at 19:03 | {
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If u expand your summation you get series $$1+0.7+0.7^2+\dots$$ as it is geometric series (you can see it here http://en.wikipedia.org/wiki/Geometric_series) $$\sum_{n=0}^{\infty}{0.7^n}=\frac{1}{1-0.7}$$
Or $S=1+x+x^2+\dots$
$xS_n=x+x^2+\dots=S-1$ now u take
$S-xS=1$
$S(1-x)=1 \implies S=\frac{1}{1-x}$
here your $x=0.7$
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The index in the sum should be $n$, not $i$. – Arturo Magidin Mar 25 '11 at 17:10
oops..i'll edit it – amul28 Mar 26 '11 at 3:54
I find the proof here lovely.
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The link here no longer seems to work. – Baby Dragon Oct 6 '13 at 1:19
@BabyDragon Here is the updated link. – David Mitra Oct 6 '13 at 9:12
## protected by AryabhataMar 31 '11 at 17:18
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# A combinatorial identity
I hope this is a suitable MO question. In a research project, my collaborator and I came across some combinatorial expressions. I used my computer to test a few numbers and the pattern was suggesting the following equation for fixed integers $K\geq n>0$.
$$\dfrac{K!}{n!K^{K-n}}\sum\limits_{ \begin{subarray}{c} k_1+\dotsb+k_{n}=K \\ k_i \geq 1 \end{subarray}} \prod\limits_{i=1}^n \dfrac{k_i^{k_i-2}}{(k_i-1)!}=\displaystyle {K-1\choose n-1}.$$
We tried to think of a proof but failed. One can probably move these $K!, n!$ to the right and rewrite the RHS, or maybe move $K!$ into the summation to form combinatorial numbers like $K\choose k_1,k_2,\dotsc,k_n$. We don't know which is better.
The questions are:
1. Anyone knows a proof for this identity?
2. In fact the expression that appears in our work is $\sum\limits_{ \begin{subarray}{c} k_1+\dotsb+k_{n}=K \\ k_i \geq 1 \end{subarray}} \sigma_p(k_1,\dotsc,k_n) \prod\limits_{i=1}^n \dfrac{k_i^{k_i-2}}{(k_i-1)!}$, where $p$ is a fixed integer and $\sigma_p(\dotsc)$ is the $p$-th elementary symmetric polynomial. The equation in the beginning simplifies this expression for $p=0,1$. Is there a similar identity for general $p$?
----------Update----------
Question 2 is perhaps too vague, and I'd like to make it a bit more specific. Probably I should have written this down in the beginning, but I feared this is too long and unmotivated. But after seeing people's skills, I'm very tempted to leave it here in case somebody has remarks. In fact, question 2 partly comes from the effort to find a proof for the following (verified by computer).
$$\frac{1}{K!} \prod_{r=1}^{K} (r+1 -x)= \sum_{n=1}^K \frac{(-1)^n}{n!} \left[ \sum_{p=0}^n K^{n-p} \prod_{r=1}^p (x +r-4) \left( \sum\limits_{ \begin{subarray}{c} k_1+\dotsb+k_{n}=K \\ k_i \geq 1 \end{subarray}} \sigma_p(k_1,\dotsc,k_{n}) \prod\limits_{i=1}^n \dfrac{k_i^{k_i-2}}{(k_i-1)!} \right) \right],$$ Where $x$ is a fixed number (in our case, an integer). | {
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• Combinatorial reformulation: consider all the trees $T$ on $\{0,1,\dots,K\}$ for which the vertices $1,2,\dots,p$ are in different components of $T\setminus \{0\}$. For each such a tree take a summand $(-K)^{\deg(0)-p}$. The sum equals $(K-2)!/(p-2)!$ for $p\geqslant 2$ and 0 for $p=0,1$. – Fedor Petrov Aug 15 '17 at 9:35
This is the answer to the first question, I wrote a long answer to Question 2 as a separate answer.
Note that $A:=\sum_{k_i>0,k_1+\dots+k_n=K}\frac{K!}{n!k_1!\dots k_n!} \prod k_i^{k_i-1}$ is a number of forests on the ground set $\{1,2,\dots,K\}$ having exactly $n$ connected components and with a marked vertex in each component ($k_i$ correspond to the sizes of components.) Add a vertex 0 and join it with the marked vertices. Then we have to count the number of trees on $\{0,1,\dots,K\}$ in which $0$ has degree $n$. Remember that the sum of $z_0^{d_1-1}\dots z_K^{d_K-1}$ over all trees on $\{0,\dots,K\}$, where $d_i$ is degree of $i$, equals $(z_0+\dots+z_K)^{K-1}$. Substitute $z_{1}=\dots=z_K=1$ and take a coefficient of $z_0^{n-1}$. It is $\binom{K-1}{n-1}K^{K-n}$.
• Thank you! This is a really cool argument, especially for me who doesn't have a lot of experiences in graph theory. I will mark this as the correct answer, but of course any further ideas and comments about question 2 are welcomed as well. – Honglu Aug 13 '17 at 18:41
• You need to talk to someone who's familiar with Volume 2 of Stanley. (This is more algebraic combinatorics than graph theory.) – Alexander Woo Aug 13 '17 at 23:10
• Sorry about my ignorance and thank you for pointing out the right words! I am thinking about asking around, and your comment definitely helps. – Honglu Aug 13 '17 at 23:58
Here's another proof. We first rewrite the identity (by setting $k_i=j_i+1$) as $$\sum_{j_1+\cdots +j_n=K-n}\prod_{i=1}^n \frac{(j_i+1)^{j_i-1}}{j_i!} = n\frac{K^{K-n-1}}{(K-n)!}. \tag{1}$$ | {
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Let $F(x)$ be the formal power series satisfying $F(x)= e^{xF(x)}$. It is well known (and easily proved, e.g., by Lagrange inversion) that $$F(x)^n = \sum_{j=0}^\infty n(j+n)^{j-1}\frac{x^j}{j!}.\tag{2}$$ In particular, $$F(x) = \sum_{j=0}^\infty (j+1)^{j-1}\frac{x^j}{j!}.$$ So the left side of $(1)$ is equal to the coefficient of $x^{K-n}$ in $F(x)^{n}$, which by $(2)$ is equal to the right side of $(1)$.
• Cool! I actually updated my question to include a more general identity that I want to prove. I kept thinking there is some generating function lurking behind, and I'm just looking at the coefficients. Any thoughts will be greatly appreciated. – Honglu Aug 14 '17 at 16:04
Here is a generating-function proof of your conjectured identity (and an answer to question 2).
The main ingredient is a formula for the appearing symmetric sums.
Let $T(z)$ (the tree function'') be the formal power series satisfying $T(z)=z\,e^{T(z)}$.
If $F$ is a formal power series the coefficients of $G(z):=F(T(z))$ are given by (Lagrange inversion) $$[z^0]G(z)=[z^0] F(z) \mbox{ , } [z^k]G(z)=\tfrac{1}{k} [y^{k-1}] F^\prime(y)\,e^{ky} =[y^k](1-y)F(y)\,e^{ky}\mbox{ for } k\geq 1\;.$$ In particular (as is well known) $$T(z)=\sum_{n\geq 1}\frac{n^{n-1}}{n!}z^n \;\mbox{ and }\; \frac{T(z)}{1-T(z)}=\sum_{n\geq 1}\frac{n^{n}}{n!}z^n$$ Thus $T(z)\left(1+\frac{t}{1-T(z)}\right)=\sum_{n\geq 1} \frac{(1+tn)n^{n-1}}{n!}$. Therefore \begin{align*}S_{p,n}(K):&=\sum_{{k_1+\ldots +k_n=K \atop k_i\geq 1}} \sigma_p(k_1,\ldots,k_n)\prod_{i=1}^n \frac{k_i^{k_i-1}}{k_i!}\\ &=[t^p]\sum_{k_1+\ldots +k_n=K \atop k_i\geq 1} \prod_{i=1}^n \frac{(1+tk_i) k_i^{k_i-1}}{k_i!}\\ &=[t^p z^K]\, T(z)^n \left(1+\frac{t}{1-T(z)}\right)^n \end{align*} and $$S_{p,n}(K)={n \choose p} [z^K]\frac {T(z)^n}{(1-T(z))^p}={n \choose p}[y^K]\,y^n\, \frac{(1-y)}{(1-y)^p}\,e^{Ky}\;\;\;\;\;(*)$$ | {
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Now consider the sum ($m:=x-4$) $$R(K,m):=\sum_{n=0}^K\frac{(-1)^n}{n!}\bigg[\sum_{p=0}^n K^{n-p}\,\left(\prod_{r=1}^p (m+r)\right) S_{n,p}(K)\bigg]$$ Since $K\geq 1$ the sum remains unchanged if we start the summation at $n=0$ (all $S_{0,p}(K)$ are $0$). Using that and $(*)$ gives \begin{align*} R(K,m):&=\sum_{n=0}^K\frac{(-1)^n}{n!}\bigg[\sum_{p=0}^n K^{n-p}\,\left(\prod_{r=1}^p (m+r)\right) S_{n,p}(K)\bigg]\\ &=[y^K]\,(1-y) \sum_{n=0}^K\frac{(-1)^n}{n!}\sum_{p=0}^n K^{n-p}\,p!{m+p \choose p}{n \choose p}\frac{y^n}{(1-y)^p}\,e^{Ky}\\ &=[y^K]\,(1-y) \sum_{p=0}^K\sum_{n=p}^K\frac{(-1)^n}{(n-p)!} K^{n-p}{m+p \choose p}\frac{y^n}{(1-y)^p}\,e^{Ky}\\ &=[y^K]\,(1-y) \sum_{p=0}^K {m+p \choose p}\frac{(-1)^p y^p}{(1-y)^p}\bigg[\sum_{n=p}^K\frac{(-1)^{n-p}}{(n-p)!} K^{n-p}y^{n-p}\,e^{Ky}\bigg]\\ &=[y^K]\,(1-y) \sum_{p=0}^K {m+p \choose p}\frac{(-1)^p y^p}{(1-y)^p}\bigg[1 +\mathcal{O}(y^{K-p+1})\bigg]\\ \end{align*} where $\mathcal{O}(y^{K-p+1})$ denotes a formal power series which is a multiple of $y^{K-p+1}$.
Taking into account the respective factors $y^p$ the terms in these series do not contribute to the coefficient $[y^K]$. Therefore
\begin{align*} R(K,m)&=[y^K]\,(1-y) \sum_{p=0}^K {m+p \choose p}\frac{(-1)^p y^p}{(1-y)^p}\\ &=[y^K]\,(1-y) \sum_{p\geq 0} {m+p \choose p}\frac{(-1)^p y^p}{(1-y)^p}\\ &=[y^K]\,(1-y) \left(\frac{1}{1+\tfrac{y}{1-y}}\right)^{m+1}\\ &=[y^K]\,(1-y)^{m+2}\\ &=(-1)^K\,{m+2 \choose K},\,\mbox{ as desired }. \end{align*} | {
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• Nice! I really learned a lot from all the answers and appreciate everybody's effort. In particular, this generating function method seems to have some other applications in our project. I got another quick question. Let $H_n=\sum\limits_{k=1}^n 1/k$. Do you know whether we can similarly describe the formal series $\sum_{n\geq 1} \dfrac{H_nn^n}{n!}z^n$ using functional equations just like your $T(z)$ and others? – Honglu Aug 15 '17 at 22:33
• Thank you. (1) The formal series $\sum_{n\geq 1}\frac{n^n}{n!}p(n) z^n$ can be expressed as a rational function of $T(z)$ if $p$ is a polynomial in $n$ and $\tfrac{1}{n}$. My first guess is that it will not be easy to treat the case $p(n)=H_n$ via Lagrange inversion. (2) $T(z)$ is certainly not "my" function. $T(z)=-W(-z)$ where $W$ is "Lambert's W-function", and $T(z)=z e^{T(z)}$ goes back to Eisenstein. – esg Aug 16 '17 at 17:57
• Would you mind telling me your real name by email? Because if we decide to post anything about this work in the future, we will acknowledge you (also Fedor and other people). I just temporarily added my email in my MO profile. We are still working on an ultimate combinatorial expression that includes all my questions as special cases. It's too long to post in a comment, but if you are interested, I will be glad to send it to you by email as well. Of course interests from other people will also be welcomed, just let me know in the comment or send me an email. – Honglu Aug 24 '17 at 16:09
Here is the answer to Question 2. It may be probably simplified. | {
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Here is the answer to Question 2. It may be probably simplified.
Denote $y=3-x$, then we rewrite your identity as $$\binom{y+K-2}K=\frac{(y-1)y(y+1)\dots (y+K-2)}{K!}=c_0\binom{y}0+c_1\binom{y}1+\dots+c_K\binom{y}K,$$ where $$c_p=p!\sum_{n=p}^K(-K)^{n-p}\frac1{n!}\sum\limits_{ \begin{subarray}{c} k_1+\dotsb+k_{n}=K \\ k_i \geq 1 \end{subarray}} \sigma_p(k_1,\dotsc,k_{n}) \prod\limits_{i=1}^n \dfrac{k_i^{k_i-2}}{(k_i-1)!}.$$ On the other hand, by Vandermonde--Chu identity we have $$\binom{y+K-2}K=\sum_{i=2}^{K}\binom{y}i\binom{K-2}{K-i},$$ so your identity is equivalent to the formula $$\sum_{n=p}^K(-K)^{n-p}\frac{K!}{n!}\sum\limits_{ \begin{subarray}{c} k_1+\dotsb+k_{n}=K \\ k_i \geq 1 \end{subarray}} \sigma_p(k_1,\dotsc,k_{n}) \prod\limits_{i=1}^n \dfrac{k_i^{k_i-2}}{(k_i-1)!}=\frac{K!}{p!}\binom{K-2}{K-p},$$ I multiplied both parts by $K!/p!$. Note that $$\frac{K!}{n!}\sum\limits_{ \begin{subarray}{c} k_1+\dotsb+k_{n}=K \\ k_i \geq 1 \end{subarray}} \sigma_p(k_1,\dotsc,k_{n}) \prod\limits_{i=1}^n \dfrac{k_i^{k_i-2}}{(k_i-1)!}$$ is a number of the trees $T$ on $\{0,1,\dots,K\}$ such that degree of 0 equals $n$ and $p$ vertices in different components of $T\setminus\{0\}$ are marked. Indeed, if these components $A_1,\dots,A_n$ are enumerated (this corresponds to the multiple $n!$) and $i$-th component $A_i$ has $k_i$ vertices, then we have $\frac{K!}{k_1!\dots k_n!}$ ways to choose $A_i$, $\sigma_p(k_1,\dotsc,k_{n})$ ways to mark $p$ vertices in different components, $k_i^{k_i-1}$ ways to make a tree on $A_i$ and choose a vertex in $A_i$ joined with 0.
Note that each (out of $\binom{K}p$ sets) set of $p$ marked vertices makes the same contribution to the sum. So, we may suppose that the marked set is $\{1,2,\dots,p\}$ and we have to prove that the sum of $(-K)^{n-p}$ over admissible trees (where the tree $T$ is admissible if $1,2,\dots,p$ are in different components of $T\setminus \{0\}$) equals $\frac1{\binom{K}p}\frac{K!}{p!}\binom{K-2}{K-p}=(p-1)p\dots (K-2)$. | {
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We start to prove this from the case $p=0$, $p=1$, where the restriction that $1,2,\dots,p$ are in different components of $T\setminus \{0\}$ disappears. Then the sum $z_0^{n-1}z_1^{d_1-1}\dots z_K^{d_K-1}$, $d_i=\deg(i)$, over all trees on $\{0,\dots,K\}$ equals, as is well known and easy to prove, to $(z_0+\dots+z_K)^{K-1}$. Substituting $z_0=-K$, $z_1=\dots=z_K=1$ we get the result.
Now we deal with the more involved case $p\geqslant 2$. Denote $K=p+m$ and consider the variables $z_0,z_1,\dots,z_p,z_{p+1},\dots$ (infinitely many for simplicity of notations). Denote $s=z_0+z_1+\dots$, write $\sigma_i$ for the $i$-th elementary symmetric polynomial of $z_{p+1},z_{p+2},\dots$. Denote $\varphi_0=1$, $\varphi_m=s\varphi_{m-1}+(p-1)p\dots (p+m-2)\sigma_m$ for $m\geqslant 1$. I claim that the sum of $z_0^{n-p}z_1^{d_1-1}\dots z_{p+m}^{d_{p+m}-1}$ over all admissible trees equals $\varphi_m(z_0,z_1,\dots,z_{p+m},0,0,\dots)$.
Note that this implies our claim, as follows from the substitution $z_0=-K=-p-m,z_1=\dots=z_{p+m}=1$.
The proof is on induction in $m$. Base $m=0$ is clear. For the induction step, look at coefficients of any specific monomial $z_0^{n-p}z_1^{d_1-1}\dots z_{p+m}^{d_{p+m}-1}$. Consider two cases:
1) $d_i=1$ for a certain index $i\in \{p+1,\dots,p+m\}$, without loss of generality $i=p+m$. This corresponds to the case when $p+m$ has degree 1, such a vertex may be joined with any of other vertices, and removing corresponding edge we get a tree (it remains admissible) on $\{0,1,\dots,K-1\}$. This corresponds to the summand $s\varphi_{m-1}$: namely, $z_j\varphi_{m-1}$ corresponds to the edge between $p+m$ and $j$; $j=0,1,\dots,p+m-1$. | {
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2) $d_{p+1},\dots,d_{p+m}$ are greater than 1. Then they are all equal to 2, since the degree of the whole monomial equals $m$. In this case there are $p(p+1)\dots (p+m-1)$ admissible trees (well, they are all admissible for such a choice of degrees and we may either apply the above formula for all trees, or prove it by induction, or as you wish). It remains to prove that the coefficient of $z_{p+1}\dots z_{p+m}$ in the function $\varphi_m$ equals $p(p+1)\dots (p+m-1)$. Since $\varphi_m=s\varphi_{m-1}+(p-1)p\dots (p+m-2)\sigma_m$, it is equivalent to proving that the coefficient of $z_{p+1}\dots z_{p+m}$ in $s\varphi_{m-1}$ equals $p(p+1)\dots (p+m-1)-(p-1)p\dots (p+m-2)=mp(p+1)\dots(p+m-2)$. We should take some $z_j$, $p+1\leqslant j\leqslant p+m$, from the multiple $s=\sum z_i$, and for each choice of $j$ we have a coefficient of $z_j^{-1}\cdot z_{p+1}\dots z_{p+m}$ in $\varphi_{m-1}$ equal to $p(p+1)\dots(p+m-2)$ - by induction (base $m-1=0$ is clear).
• Not sure whether you get notifications from comments in other answers or not. To be sure I just repeat some messages here. I really appreciate all your answers here. Ultimately we want to simplify a bigger expression that includes the RHS of my question 2 as a special case. It's pretty long and currently still over my head. I don't know if you have the time and the interest to take a look. But in case you do, you are welcomed to send me an email (in my profile). – Honglu Aug 24 '17 at 19:55 | {
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# power, .^
Element-wise power
## Description
example
C = A.^B raises each element of A to the corresponding powers in B. The sizes of A and B must be the same or be compatible.
If the sizes of A and B are compatible, then the two arrays implicitly expand to match each other. For example, if one of A or B is a scalar, then the scalar is combined with each element of the other array. Also, vectors with different orientations (one row vector and one column vector) implicitly expand to form a matrix.
C = power(A,B) is an alternate way to execute A.^B, but is rarely used. It enables operator overloading for classes.
## Examples
collapse all
Create a vector, A, and square each element.
A = 1:5;
C = A.^2
C = 1×5
1 4 9 16 25
Create a matrix, A, and take the inverse of each element.
A = [1 2 3; 4 5 6; 7 8 9];
C = A.^-1
C = 3×3
1.0000 0.5000 0.3333
0.2500 0.2000 0.1667
0.1429 0.1250 0.1111
An inversion of the elements is not equal to the inverse of the matrix, which is instead written A^-1 or inv(A).
Create a 1-by-2 row vector and a 3-by-1 column vector and raise the row vector to the power of the column vector.
a = [2 3];
b = (1:3)';
a.^b
ans = 3×2
2 3
4 9
8 27
The result is a 3-by-2 matrix, where each (i,j) element in the matrix is equal to a(j) .^ b(i): | {
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The result is a 3-by-2 matrix, where each (i,j) element in the matrix is equal to a(j) .^ b(i):
$\mathit{a}=\left[{\mathit{a}}_{1}\text{\hspace{0.17em}}{\mathit{a}}_{2}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathit{b}=\left[\begin{array}{c}{\mathit{b}}_{1}\\ {\mathit{b}}_{2}\\ {\mathit{b}}_{3}\end{array}\right],\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\text{\hspace{0.17em}}\mathit{a}\text{\hspace{0.17em}}.ˆ\text{\hspace{0.17em}}\mathit{b}=\left[\begin{array}{cc}{{\mathit{a}}_{1}}^{{\mathit{b}}_{1}}& {{\mathit{a}}_{2}}^{{\mathit{b}}_{1}}\\ {{\mathit{a}}_{1}}^{{\mathit{b}}_{2}}& {{\mathit{a}}_{2}}^{{\mathit{b}}_{2}}\\ {{\mathit{a}}_{1}}^{{\mathit{b}}_{3}}& {{\mathit{a}}_{2}}^{{\mathit{b}}_{3}}\end{array}\right].$
Calculate the roots of -1 to the 1/3 power.
A = -1;
B = 1/3;
C = A.^B
C = 0.5000 + 0.8660i
For negative base A and noninteger B, the power function returns complex results.
Use the nthroot function to obtain the real roots.
C = nthroot(A,3)
C = -1
## Input Arguments
collapse all
Operands, specified as scalars, vectors, matrices, or multidimensional arrays. A and B must either be the same size or have sizes that are compatible (for example, A is an M-by-N matrix and B is a scalar or 1-by-N row vector). For more information, see Compatible Array Sizes for Basic Operations.
• Operands with an integer data type cannot be complex.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical | char
Complex Number Support: Yes
collapse all
### IEEE Compliance
For real inputs, power has a few behaviors that differ from those recommended in the IEEE®-754 Standard.
MATLAB® IEEE
power(1,NaN)
NaN
1
power(NaN,0) | {
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MATLAB® IEEE
power(1,NaN)
NaN
1
power(NaN,0)
NaN
1
## Compatibility Considerations
expand all
Behavior changed in R2016b
## Extended Capabilities
### Topics
Introduced before R2006a | {
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Sum Of Sequence Calculator | {
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Use this calculator to determine the future value of your savings and lump sum. In geometric progressions where |r| < 1 (in other words where r is less than 1 and greater than –1), the sum of the sequence as n tends to infinity approaches a value. This calculator will find the infinite sum of arithmetic, geometric, power, and binomial series, as well as the partial sum, with steps shown (if possible). The Arithmetic series of finite number is the addition of numbers and the sequence that is generally followed include – (a, a + d, a + 2d, …. Approach : Read input number asking for length of the list using input() or raw_input(). Sum of 32-bit integer quantities is not computed by using 64-bit results, and overflow can occur for the LINQ to SQL translation of Sum. This free number sequence calculator can determine the terms (as well as the sum of all terms) of an arithmetic, geometric, or Fibonacci sequence. Sequences and summations CS 441 Discrete mathematics for CS M. Objects might be numbers or letters. Set Sum = 0. The limit of a sequence is the value the sequence approaches as the number of terms goes to infinity. Partial Sums of an Arithmetic Sequence. All term in the sequence meet a specific logical rule which needs to be recognised in order to find the missing terms. Because a series consists of evenly-spaced numbers, the median and mean (average) of the series will be the same. arithmetic series. This way, each term can be expressed by this equation: xₐ = xₐ₋₁ + xₐ₋₂. ): A series of numbers in which ratio of any two consecutive numbers is always a same number that is constant. Calculates the sum of the infinite geometric series. Arithmetic Sequences and Sums Sequence. The first of the examples provided above is the sum of seven whole numbers, while the latter is the sum of the first seven square numbers. We start with the formula for PV of a future value (FV) single lump sum at time n and interest rate i,. lets say you want to sum the numbers from 1 to 100 | {
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single lump sum at time n and interest rate i,. lets say you want to sum the numbers from 1 to 100 all you have to do is to calculate this (100)x(101) /2. What Is The Formula For Calculating Arithmetic Sequence? If the initial term of an arithmetic sequence is a 1 and the common difference of successive members is d , then the nth term of the sequence is given by:. The sequence begins 1, 1, 2, 3, 5, and each succeeding term is the sum of the previous two terms. If only a single number for value1 is supplied, SUM returns value1. Example 1: Find the sum of squares of the numbers from 0 to 5000. I want to calculate sum of the average daily value off each product. Work out: a. Sample Output: Input The Value For Nth Term: 5 1*1 = 1 2*2 = 4 3*3 = 9 4*4 = 16 5*5 = 25 The Sum Of The Above Series Is: 55 ASAP🙏 This problem has been solved!. skipna bool, default True. The Corbettmaths video tutorial on finding the nth term for a fractional sequence. In this case, you have the sequence. Note that a series is an indicated sum of the terms of a sequence!! In this section, we work only with finite series and the related sums. P Class 11 Engineering - Sum of n Terms of G. The formula for the n-th term of a quadratic sequence is explained here. And how many pairs do we have? Well, we have 2 equal rows, we must have n/2 pairs. The sequence of terms that are common to both sequences will be an arithmetic sequence with a common difference of 28, which is the LCM of 4 and 7. Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. Add Number to Sum and set equal to Sum. This calculator will find the infinite sum of arithmetic, geometric, power, and binomial series, as well as the partial sum, with steps shown (if possible). What is Sum of the Series in Math. The arithmetic sequence calculator uses arithmetic sequence formula to find sequence of any property. The problem is that the number of cells to be summed will vary; for one run of the macro it could be | {
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The problem is that the number of cells to be summed will vary; for one run of the macro it could be 100 cells, while on the next it could be 300 and on the third only 25. std ([ddof]) Calculate window standard deviation. Sequence calculator allows to calculate online the terms of the sequence whose index is between two limits. Geometric Progression (G. Sequence: By a sequence, we mean an arrangement of number in definite order according to some rules. Build your own widget. Improve your math knowledge with free questions in "Find the sum of an arithmetic series" and thousands of other math skills. The sum of a geometric series is indeed an interesting place to start this discussion. You also generate a sum in each step, which you reuse in the next step. Home; Java Tutorial; Language; Data Type; Operators; Statement Control. Find the sum of first n terms of the series (i) 3 +33 +333 +. Then we update the value of variable sum as - sum = sum + c = 0 + 0 = 0. To concatenate a series of iterables, consider using itertools. help me how to calculate the sum of a series in Matlab. The Fibonacci sequence typically has first two terms equal to x₀ = 0 and x₁ = 1. In other words, how to take the value of a cell located in one worksheet and add it to the value of another cell located in another worksheet to come up with the total of the respective cells. The sequence begins 1, 1, 2, 3, 5, and each succeeding term is the sum of the previous two terms. Calculate the sum of series 1^2+2^2+3^2+4^2++n^2 (n>0) using the both for and while loop structure. Subtract the smaller fraction from the larger fraction to calculate the. Sum of 32-bit integer quantities is not computed by using 64-bit results, and overflow can occur for the LINQ to SQL translation of Sum. The partial sum approach of course involves a "trick" as well -- finding an expression for the dang partial sum. To find sum of your series, you need to choose the series variable, lower and upper bounds and also input the | {
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of your series, you need to choose the series variable, lower and upper bounds and also input the expression for n-th term of the series. Example 2: Find the sum of squares of the first 100 numbers of the form prime minus one. This calculator for to calculating the sum of a series is taken from Wolfram Alpha LLC. As n tends to infinity, S n tends to The sum to infinity for an arithmetic series is undefined. Leonhard Euler was able to calculate the exact sum of the p-series with p 2: 2-2 32 42 Use this fact to find the sum of each series: 2 32 so. Recently, mandatory vote-by-mail has received a great deal of attention as a means of administering elections in the United States. 01 then we use the formula for the sum of the infinite geometric series S oo = a 1 / (1 - r),. Using for loop. Program description:- Find sum of series 1² + 2² + 3² + 4² + 5² +…. com presented a large number of task in mathematics that you can solve online free of charge on a variety of topics: calculation of integrals and derivatives, finding the sum of the series, the solution of differential equations, etc. You may find functions such as sum, ones, or dot useful. In other words, how to take the value of a cell located in one worksheet and add it to the value of another cell located in another worksheet to come up with the total of the respective cells. sum (axis = None, skipna = None, level = None, numeric_only = None, min_count = 0, ** kwargs) [source] ¶ Return the sum of the values for the requested axis. If the value of the sum (in the limiting sense) exists, then they say that the series. Calculate the first term. Guidelines to use the calculator If you select a n, n is the nth term of the sequence If you select S n, n is the first n term of the sequence For more information on how to find the common difference or sum, see this lesson arithmetic sequence. Our sum of series calculator or arithmetic series calculator is an online tool which you can find on Google. is 1 and any term is | {
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or arithmetic series calculator is an online tool which you can find on Google. is 1 and any term is equal to the sum of all the succeeding terms. It is used to achieve the gas–liquid or liquid–liquid two-phase separation. NPV = Sum CF* ((1+2%)/(1+D))^N When N is infinite, after simplification, NPV = CF * 1 / (1 - r) where r = (1+2%)/(1+D)--A+ V. In general, one does not expect to be able to calculate an infinite sum exactly. In this case, you have the sequence. But there are some series. By definition, the first two numbers in the Fibonacci sequence are either 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two. Java program to calculate the sum of N numbers using arrays, recursion, static method, using while loop. Each number in series is called as Fibonacci number. Leonhard Euler continued this study and in the process solved many. Finding a Rule for a Sequence [07/24/2003] What is the next number in this sequence? 1, 3, 11, 67, ? Finding Sum Formula using Sequences of Differences [06/28/1998] Finding a formula for the sum of the first n fourth powers using sequences of differences. 02 Line 10 Column 10 "The Sum of the above Series for term "s is : ". Find the sum of 35 terms of an arithmetic series of which the first term is "a" and the fifteenth term is "9a. It is said that when Gauss was ten. We start with the formula for PV of a future value (FV) single lump sum at time n and interest rate i,. I evaluated the partial sum through a calculator, my answer was -19/30, or -. Sample Output: Input The Value For Nth Term: 5 1*1 = 1 2*2 = 4 3*3 = 9 4*4 = 16 5*5 = 25 The Sum Of The Above Series Is: 55 ASAP🙏 This problem has been solved!. After discussing the above two examples one will wonder if any sequence has the same faith (meaning, it gets closer to a number). Sum of this series is calculated by the formula : sum=(n*(n+1))/2. It will also check whether the series converges. So, the | {
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by the formula : sum=(n*(n+1))/2. It will also check whether the series converges. So, the given series is an arithmetic progression whose common difference is d = 2. An arithmetic sequence is one in which the difference between successive members is a constant. demonstration: form input vectors for 10, ten thousand, and 1*10^(many) 0 10 3 0 10 5 0 10 15 0 10000. We therefore derive the general formula for evaluating a finite arithmetic series. The limit of a sequence is the value the sequence approaches as the number of terms goes to infinity. Geometric Progression, Series & Sums Introduction. By applying this calculator for Arithmetic & Geometric Sequences, the n-th term and the sum of the first n terms in a sequence can be accurately obtained. Sequence calculator allows to calculate online the terms of the sequence whose index is between two limits. The T-junction is a novel type of separator used in the petroleum and gas industry. What two things do you need to know to find the sum of an infinite geometric series? Find the sum of the infinite geometric series. asked by ryan on March 15, 2017; Algebra. Explain why or why not. Please help. The two sequences are placed in two consecu- tive rows. The sums we have looked at so far are finite sums with finite upper and lower limits. I know the inequality of [integral from n+1 to infinity of An] < [Total sum - Partial sum] < [integral from n to infinity of An]. Thanks for your help. You need to know both of these numbers in order to calculate the sum of the arithmetic sequence. 1 - Enter the first term A1 in the sequence, the common ratio r and n n the number of terms in the sum then press enter. Hope this helps!. S = 10 1 2, a 1= 1 2 Write. +N^2 in C Programming Language. Such series appear in many areas of modern mathematics. This formula shows that a constant factor in the summands can be taken out of the sum. Free Summation Calculator. However, when the series has an infinite number of terms the summation is more | {
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Summation Calculator. However, when the series has an infinite number of terms the summation is more complicated and the series may or may not have a finite. Let the variable equal the first term in the sequence, and equal the last term in the sequence. Observe that for the geometric series to converge, we need that $$|r| 1$$. Excel's Calculation Process. Any sequence that has a common second difference is a quadratic sequence. So the second term common to both sequences is 1+28 = 29. arithmetic series. asked by ryan on March 15, 2017; Algebra. Infinite Series calculator is a free online tool that gives the summation value of the given function for the given limits. In the opposite case, one should pay the attention to the «Series convergence test» pod. Sum of series in C language 1² + 2² + 3² + 4² + 5² +…. Linear sequences A number pattern which increases (or decreases) by the same amount each time is called a linear sequence. Sequences and summations CS 441 Discrete mathematics for CS M. s n = 2 − 3(0. Get comfortable with sequences in general, and learn what arithmetic sequences are. com Task : To find the sum of all the elements in a list. Calculates the 8-bit checksum for a sequence of hexadecimal bytes. On our site OnSolver. I understand the process in calculating a simple series like this one $$\\sum_{n=0}^4 2n$$ but I do not understand the steps to calculate a series like this one $$\\sum_{n=1}^x n^2$$ I have an aw. Note that a series is an indicated sum of the terms of a sequence!! In this section, we work only with finite series and the related sums. Learn Java by Examples: Write a program to calculate and print the sum of the following series: Sum(x) = 2 – 4 + 6 – 8 + 10 - 12 … - 100Learn Java by examples. See full list on mathsisfun. The free tool below will allow you to calculate the summation of an expression. S = 12, a 1= 2 8. This Arithmetic Sequence Calculator is used to calculate the nth term and the sum of the first n terms of an arithmetic | {
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Calculator is used to calculate the nth term and the sum of the first n terms of an arithmetic sequence. First simplify the expression, X! receives great quick! x/x! = x/(x * (x-a million) * (x-2) *a million) hence x/x! = a million/(x-a million)! for x >= 2 and a million for x =a million Then note that that's the same to the enlargement of the Taylor series for e^x as a million + x + x^2/2! + x^3/3! +x^4/4! for x = a million. The sequence of partial sums of a series sometimes tends to a real limit. If you're behind a web filter, please make sure that the domains *. Free series convergence calculator - test infinite series for convergence step-by-step This website uses cookies to ensure you get the best experience. This is a free javascript calculator. Leonhard Euler continued this study and in the process solved many. \$1000000007\$ is a prime number, so division by \$2^L\$ modulo 1000000007 is a multiplication by a multiplicative inverse of 2 modulo 1000000007 (which is trivial to find) to the same power. 99805 Time Complexity: O(n). In general, one does not expect to be able to calculate an infinite sum exactly. As the top row increases, the bottom row decreases, so the sum stays the same. Infinite Geometric Series Calculator is a free online tool that displays the sum of the infinite geometric sequence. The T-junction is a novel type of separator used in the petroleum and gas industry. Sum definition is - an indefinite or specified amount of money. The first of the examples provided above is the sum of seven whole numbers, while the latter is the sum of the first seven square numbers. The first sum in the numerator is the sum of squared residuals for the first model (e. It is used to achieve the gas–liquid or liquid–liquid two-phase separation. On a higher level, if we assess a succession of numbers, x 1 , x 2 , x 3 ,. This document explains how to calculate the sum or total when working with cell data located in multiple worksheets. Explain why or why not. | {
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the sum or total when working with cell data located in multiple worksheets. Explain why or why not. Finding a general expression for a partial sum by induction and then finding the limit of this partial sum is a perfectly valid technique. Question: Find the sum of (1/2) +(1/6) + (1/12) + + (1/9900) without using a calculator. Improve your math knowledge with free questions in "Find the sum of an arithmetic series" and thousands of other math skills. Get comfortable with sequences in general, and learn what arithmetic sequences are. var ([ddof]) Calculate unbiased window variance. 2, ALPHA ([I], 1, 1 0) EXE, and you should get an answer of 9. The z-transform of the unit pulse, = 1. Learn Java by Examples: Write a program to calculate and print the sum of the following series: Sum(x) = x/2 + x/5 + x/8 + … + x/100. BYJU'S online infinite geometric series calculator tool makes the calculation faster, and it displays the sum in a fraction of seconds. Sequences and summations CS 441 Discrete mathematics for CS M. Geometric Progression (G. Sum of Natural, Odd or Even Numbers Series Calculator getcalc. Java program to calculate the sum of GP series. Just enter the expression to the right of the summation symbol (capital sigma, Σ) and then the appropriate ranges above and below the symbol, like the example provided. This summation notation calculator can sum up many types of sequencies including the well known arithmetic and geometric sequencies, so it can help you to find the terms including the nth term as well as the sum of the first n terms of virtualy any series. Series (Find the sum) A finite Geometric Series (a limited number of terms, or Partial Sum) An infinite Geometric Series, if our infinite series is. Solving mathematical problems online for free. help me how to calculate the sum of a series in Matlab. The sums we have looked at so far are finite sums with finite upper and lower limits. To evaluate the Midpoint Sum, it gets messier. For some use cases, there | {
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upper and lower limits. To evaluate the Midpoint Sum, it gets messier. For some use cases, there are good alternatives to sum(). The sum of the members of a finite arithmetic progression is called an arithmetic series. Sn = (n/2) [a + l] Sn = (n/2) [2a + (n - 1)d] a = first term, n = number of terms of the series, d = common difference and l = last term. 0 / k for k in range(1, 10001)) What this code does: the innermost part is a generator expression, which computes the elements of a series 'on the fly'. Example 1: Find the sum of squares of the numbers from 0 to 5000. For example, in the sequence 10, 15, 20, 25, 30. Observe that for the geometric series to converge, we need that $$|r| 1$$. So again, a problem about earned interest might not be a perfect example, since you can withdraw your money at any instant and not only at whole number year values. Sample Output: Input The Value For Nth Term: 5 1*1 = 1 2*2 = 4 3*3 = 9 4*4 = 16 5*5 = 25 The Sum Of The Above Series Is: 55 ASAP🙏 This problem has been solved!. 1 26) a 12 = 28. Arithmetic Sequence. \) The series becomes $$\sum\limits_{n = 0}^\infty {\large\frac{{{u^n}}}{{n!}} ormalsize}. More Practice Problems with Arithmetic Sequence Formula Direction: Read each arithmetic sequence question carefully, then answer with supporting details. Given real (or complex!) numbers aand r, X1 n=0 arn= (a 1 r if jr <1 divergent otherwise The mnemonic for the sum of a geometric series is that it’s \the rst term divided by one minus the common ratio. Another approach could be to use a trigonometric identity. The sums we have looked at so far are finite sums with finite upper and lower limits. Guidelines to use the calculator If you select a n, n is the nth term of the sequence If you select S n, n is the first n term of the sequence For more information on how to find the common difference or sum, see this lesson arithmetic sequence. the starting principal you'll need to achieve the payouts desired:. org are unblocked. Math | {
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the starting principal you'll need to achieve the payouts desired:. org are unblocked. Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. In column B, create the sequence that you are going to sum and name it seqa. A sequence is a list of numbers. After discussing the above two examples one will wonder if any sequence has the same faith (meaning, it gets closer to a number). Free Summation Calculator. [Note that the sequence of the sums S 1 (the sum of the first term), S 2 (the sum of the first two terms), S 3 (the sum of the first three terms),. The present value, PV, of a series of cash flows is the present value, at time 0, of the sum of the present values of all cash flows, CF. To determine if the series is convergent we first need to get our hands on a formula for the general term in the sequence of partial sums. Only this variable may occur in the sequence term. is 16 and sum of the next 3 terms is 128. The denominator doesn't change. This calculator for to calculating the sum of a series is taken from Wolfram Alpha LLC. How to use sum in a sentence. Because 1 is paired with 10 (our n), we can say that each column has (n+1). Alex's Arithmetic Sequence Sum Calculator is a very simple program, which allows you to get the sum of an Arithmetic Squence, it supports two types of sequences: ￭ type 1 - if you have the first. When the denominators are the same, add the numerators, or top numbers, to calculate the answer. arithmetic series. In column A, create the sequence 1, 2, 3,. ID 1001746465 1 1 1 1 23 4+5 1 There are several ways to implement this. Sum of Three Consecutive Integers calculator. I hope it is enough to get you going in the right direction. The sum of geometric series refers to the total of a given geometric sequence up to a specific point and you can calculate this using the geometric sequence solver or the geometric series calculator. Series calculator allows to calculate online the sum of the terms of the | {
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series calculator. Series calculator allows to calculate online the sum of the terms of the sequence whose index is between the lower and the upper bound. My dxp and csv file is uploaded in below. If you are struggling to understand what a geometric sequences is, don't fret! We will explain what this means in more simple terms later on and take a look at. Limit Comparison Test If lim (n-->) (a n / b n) = L, where a n, b n > 0 and L is finite and positive,. This value is equal to:. Solution: Given decimal we can write as the sum of 0. Leonhard Euler was able to calculate the exact sum of the p-series with p 2: 2-2 32 42 Use this fact to find the sum of each series: 2 32 so. Notice \t is used to provide 8 spaces (1 tab) between two values see in the output. It should be noted, that if the calculator finds sum of the series and this value is the finity number, than this series converged. hi i just want to confirm my method of using sum of pairs is correct. Because the Fibonacci value for 20000 has 4179 decimals and it needs quite an impressive amount of processing, the maximum allowed value is 20000. 0 / k for k in range(1, 10001)) What this code does: the innermost part is a generator expression, which computes the elements of a series 'on the fly'. As running variable, which is increased by 1 in each step, i is used. Excel does not calculate cells in a fixed order, or by Row or Column. We used a sum-over-states method to calculate the dynamic polarizabilities of 6S 1/2 ground state and highly-excited (nS 1/2 and nP 3/2 ) Rydberg state of cesium atoms, and identify corresponding magic detuning for optical wavelengths in the range of 850-1950 nm. Z Transforms of Common Sequences Ele 541 Electronic Testing Unit Pulse. We call an a term of the sequence. If you're seeing this message, it means we're having trouble loading external resources on our website. Representations of N as a sum of distinct elements from special sequences D. What Is Arithmetic Sequence? is a | {
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of N as a sum of distinct elements from special sequences D. What Is Arithmetic Sequence? is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. Infinite Geometric Series Calculator is a free online tool that displays the sum of the infinite geometric sequence. If the series has a finite number of terms, it is a simple matter to find the sum of the series by adding the terms. Sequence calculator allows to calculate online the terms of the sequence whose index is between two limits. this, use your calculator and examine high powers of numbers between 1 and -1. A geometric sequence is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by r. The Corbettmaths video tutorial on finding the nth term for a fractional sequence. A series, which is not a list of terms like a sequence, is the sum of the terms in a sequence. A finite number of terms of an arithmetic sequence can be added to find their sum. Often the first numbers will be 1, but not always. If S n tends to a limit as n tends to infinity, the limit is called the sum to infinity of the series. But there are some series. I evaluated the partial sum through a calculator, my answer was -19/30, or -. Exclude NA/null. In mathematics, summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total. How to Find Sum of First n Terms When nth Term is Given ? A series whose terms are in Arithmetic progression is called Arithmetic series. Sum of this series is calculated by the formula : sum=(n*(n+1))/2. The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. ID 1001746465 1 1 1 1 23 4+5 1 There are several ways to implement this. Work out: a. day, sum((random()*5)::integer) num FROM days -- left join other tables here to get counts, I'm using random group by days. The | {
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num FROM days -- left join other tables here to get counts, I'm using random group by days. The Sum (Summation) Calculator is used to calculate the total summation of any set of numbers. P Class 11 Engineering - Sum of n Terms of G. But there are some series. Calculate sum elements of sequence: sum. Doing so means that most of the depreciation associated with an asset is recognized in the first few years of its useful life. All term in the sequence meet a specific logical rule which needs to be recognised in order to find the missing terms. Thus the message becomes: Since we are using a 3 by 3 matrix, we break the enumerated message above into a sequence of 3 by 1. The first term of an infinite G. After all, yes 1/(1-x) has an honest-to-goodness explosion to infinity at x=1, but it makes perfectly good sense at x=-1, and tells us (what my calc students. Sum of elements in a list — programminginpython. Series (Find the sum) When you know the first and last term. [ Don't peek. Recently, mandatory vote-by-mail has received a great deal of attention as a means of administering elections in the United States. Write a program in C++ to calculate the sum of the series (1*1) + (2*2) + (3*3) + (4*4) + (5*5) + + (n*n). 02 Line 10 Column 42 Pic zz9 From N Highlight. I know the inequality of [integral from n+1 to infinity of An] < [Total sum - Partial sum] < [integral from n to infinity of An]. Number sequences questions usually consist of four to seven visible numbers along with a single missing number or, depending on the sequence's complexity level, 2 or 3 missing numbers. An infinite series has an infinite number of terms. Number sequences test practice for aptitude tests and psychometric IQ tests. The sum of a sequence is known as a series, and the harmonic series is an example of an infinite series that does not converge to any limit. ${s_n} = \sum\limits_{i = 1}^n i$. Sum of: from: to: Submit: Share a link to this widget: More. Solution: A series in which each number is | {
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of: from: to: Submit: Share a link to this widget: More. Solution: A series in which each number is sum of its previous two numbers is known as Fibonacci series. mean (*args, **kwargs) Calculate the window mean of the values. 23) a 21 = −1. The program solves Riemann sums using one of four methods and displays a graph when prompted. Fourth, we recall the sum of the X 2 and subtract 240. ID 1001746465 1 1 1 1 23 4+5 1 There are several ways to implement this. To be a good Java developer is to be fluent in Java 8. Sum of this series is calculated by the formula : sum=(n*(n+1))/2. Just enter the expression to the right of the summation symbol (capital sigma, Σ) and then the appropriate ranges above and below the symbol, like the example provided. The nineteenth term. 2, ALPHA ([I], 1, 1 0) EXE, and you should get an answer of 9. An arithmetic sequence is one in which the difference between successive members is a constant. To use this calculator, simply type in your list of inputs separated by commas (ie 2,5,8,10,12,18). Print the sum. Sum of sequence calculator. Sequence calculator allows to calculate online the terms of the sequence whose index is between two limits. Geometric Progression (G. asked by ryan on March 15, 2017; Algebra. For the finite sums series calculator computes the answer quite literally, so if there is a necessity to obtain a short expression we recommend computing a parameterized sum. This website uses cookies to ensure you get the best experience. Infinite Series calculator is a free online tool that gives the summation value of the given function for the given limits. I have made this small program to help me add numbers and find the sum, it runs but no output. The nth term of a sequence is given by U n = n 2 /(n + 1). 02 Line 10 Column 58 Pic zz. Everything you want to know about Java. I know the inequality of [integral from n+1 to infinity of An] < [Total sum - Partial sum] < [integral from n to infinity of An]. Guidelines to use the | {
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of An] < [Total sum - Partial sum] < [integral from n to infinity of An]. Guidelines to use the calculator If you select a n, n is the nth term of the sequence If you select S n, n is the first n term of the sequence For more information on how to find the common difference or sum, see this lesson arithmetic sequence. P Class 11 Engineering - Sum of n Terms of G. Write a program in C++ to calculate the sum of the series (1*1) + (2*2) + (3*3) + (4*4) + (5*5) + + (n*n). The sequence of these partial sums converges to also. 78 degrees Fahrenheit. The first of the examples provided above is the sum of seven whole numbers, while the latter is the sum of the first seven square numbers. com allows you to find the sum of a series online. For help creating a sequence, see Note 1I. If the range of a sum is finite, is typically assigned a sequence of values, with being evaluated for each one. Third, we square the sum of X (45 times itself = 2025) and divide it by N (number of scores). A finite number of terms of an arithmetic sequence can be added to find their sum. The 8-bit checksum is the 2's complement of the sum off all bytes. The first of the examples provided above is the sum of seven whole numbers, while the latter is the sum of the first seven square numbers. 01 then we use the formula for the sum of the infinite geometric series S oo = a 1 / (1 - r),. Such series appear in many areas of modern mathematics. Question: Find the sum of (1/2) +(1/6) + (1/12) + + (1/9900) without using a calculator. #sum_(k=1)^nk=(n(n+1))/2# and. The amount it increases or decreases by is known as the common difference. Add the numbers and divide by (n - 1) = 6 to get 95. "Mike" wrote: > I have a series of cashflows, forecast to grow at say 2% each year and > go on indefinitely. Objects might be numbers or letters. Using the nth term. Sum of a Series: If we are able to find the general term of the series successfully, then we can also. Work out: a. Which of the pairs of events below is | {
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of the series successfully, then we can also. Work out: a. Which of the pairs of events below is dependent? Select the correct answer below: drawing a 7 and then drawing another 7 with replacement from a standard deck of cards rolling a 1 and then rolling a 6 with a standard die rolling a 3 and then rolling a 4 with. The sequence of these partial sums converges to also. Thus the message becomes: Since we are using a 3 by 3 matrix, we break the enumerated message above into a sequence of 3 by 1. An example of the sequence can be seen as follows:. Infinite Geometric Series Calculator is a free online tool that displays the sum of the infinite geometric sequence. BYJU’S online infinite series calculator tool makes the calculations faster and easier where it displays the value in a fraction of seconds. It will also check whether the series converges. The numerator above is a difference of two partial sums of the \2^n*a_{n}\ sequence. The two sequences are placed in two consecu- tive rows. This technique causes problems in several situations, however and cannot be universally relied upon. Using Wolfram Alpha 214 , put in 10 x x and you will get can be reduced to the much easier-to-calculate infinite sum 1 - x - x2 Series - A series is formed by the sum or addition of the terms in a two equations that allow easy calculation for an arithmetic series. The numerator above is a difference of two partial sums of the \2^n*a_{n}\ sequence. More Practice Problems with Arithmetic Sequence Formula Direction: Read each arithmetic sequence question carefully, then answer with supporting details. day, sum((random()*5)::integer) num FROM days -- left join other tables here to get counts, I'm using random group by days. For example, two-thirds plus four-thirds equals six-thirds. Average = Sum of terms / Number of terms. is 16 and sum of the next 3 terms is 128. Methods for Evaluating In nite Series Charles Martin March 23, 2010 Geometric Series The simplest in nite series is the | {
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In nite Series Charles Martin March 23, 2010 Geometric Series The simplest in nite series is the geometric series. The sum of the first three terms of the sequence of terms common to the two given sequences is 3 times the second term of the sequence. The sum of geometric series refers to the total of a given geometric sequence up to a specific point and you can calculate this using the geometric sequence solver or the geometric series calculator. interval is [FIRST, LAST) NB. The sum of the members of a finite arithmetic progression is called an arithmetic series. Write a program in C++ to calculate the sum of the series (1*1) + (2*2) + (3*3) + (4*4) + (5*5) + + (n*n). Alternatively – put the first few terms of your sequence into WolframAlpha. Sample Output: Input The Value For Nth Term: 5 1*1 = 1 2*2 = 4 3*3 = 9 4*4 = 16 5*5 = 25 The Sum Of The Above Series Is: 55 ASAP🙏 This problem has been solved!. Next: Write a program in C++ to find the sum of series 1 - X^2/2! + X^4/4!- upto nth term. That's numberwang!. Our summation calculator can easily calculate the sum of any numbers you input. Number sequences test practice for aptitude tests and psychometric IQ tests. sin105 degrees B. the starting principal you'll need to achieve the payouts desired:. , x k , we can record the sum of these numbers in the following way:. Examples: SAS Statements. Lets say i have three sequences. Notice, that if x = 1 then, in the series, we are simply addi ng up an infinite number of 1's and of course the sum goes to infinity. Please help. An infinite series has an infinite number of terms. Beware: people often confuse the terms ‘sequence’ and ‘series’. An Efficient solution to solve the sum of geometric series where first term is a and common ration is r is by the formula :-sum of series = a(1 – r n)/(1 – r). Enter 2 numbers to add and press the = button to get the sum result. #a_n = (3/2)^n# Which means that #n#-th term is generates by raising #3/2# to the #n#-th power. Find the | {
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= (3/2)^n# Which means that #n#-th term is generates by raising #3/2# to the #n#-th power. Find the series. Sequences and summations CS 441 Discrete mathematics for CS M. [Note that the sequence of the sums S 1 (the sum of the first term), S 2 (the sum of the first two terms), S 3 (the sum of the first three terms),. That's numberwang!. This free number sequence calculator can determine the terms (as well as the sum of all terms) of an arithmetic, geometric, or Fibonacci sequence. +N^2 in C Programming Language. In other words, if you keep adding together the terms of the sequence forever, you will get a finite value. Products Classroom Activities Graphing Calculator Scientific Calculator Four Function Calculator Matrix Calculator Test Practice Geometry Tool. In other words, how to take the value of a cell located in one worksheet and add it to the value of another cell located in another worksheet to come up with the total of the respective cells. Series calculator allows to calculate online the sum of the terms of the sequence whose index is between the lower and the upper bound. sum_arithmetic_series is more general than required. what could be the problem? // Program that uses a switch statement. If this happens, we say that this limit is the sum of the series. P Class 11 Engineering - Sum of n Terms of G. All you have to do is write the first term number in the first box, the second term number in the second box, third term number in the third box and the write value of n in the fourth box after that you just have to click on the Calculate button, your result will be visible. Enter the sequence, the start value and end value from sigma notation and get a numerical sum. To add floating point values with extended precision, see math. What is the difficulty level of this exercise? Easy Medium Hard. About Sum (Summation) Calculator. Question: Find the sum of (1/2) +(1/6) + (1/12) + + (1/9900) without using a calculator. Calculate the sum of first n squares or the | {
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+(1/6) + (1/12) + + (1/9900) without using a calculator. Calculate the sum of first n squares or the sum of consecutive square numbers from n 1 2 to n 2 2. A series is a special type of sequence: a. Notice, that if x = 1 then, in the series, we are simply addi ng up an infinite number of 1's and of course the sum goes to infinity. Calculate sum elements of sequence: sum. Thanks for your help. Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. Bob has a need to use the SUM function in a macro in order to find the sum of all the values in a column. The sequence of partial sums of a series sometimes tends to a real limit. var ([ddof]) Calculate unbiased window variance. 23) a 21 = −1. The formula for the n-th term of a quadratic sequence is explained here. For example, some series don't sum monotonically to a limit -- the sum will bounce around it, and the summation's state can end up oscillatory forever. 02 Line 10 Column 42 Pic zz9 From N Highlight. skipna bool, default True. Third, we square the sum of X (45 times itself = 2025) and divide it by N (number of scores). Free Summation Calculator. +N^2 in C Programming Language. An example of the sequence can be seen as follows:. Instructions: Use this step-by-step Geometric Series Calculator, to compute the sum of an infinite geometric series by providing the initial term \(a$$ and the constant ratio $$r$$. Our sum of series calculator or arithmetic series calculator is an online tool which you can find on Google. Although SUM is specified as taking a maximum of 30 arguments, Google Sheets supports an arbitrary number of arguments for this function. About this calculator. (for example 2 series in the picture) 1 Comment. Free Arithmetic Sequences calculator - Find indices, sums and common difference step-by-step This website uses cookies to ensure you get the best experience. The sum function can be used as a series calculator, to calculate the sequence of partial sums of a series. | {
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function can be used as a series calculator, to calculate the sequence of partial sums of a series. Set Number = 0. The sums we have looked at so far are finite sums with finite upper and lower limits. The Arithmetic series of finite number is the addition of numbers and the sequence that is generally followed include – (a, a + d, a + 2d, …. day, sum((random()*5)::integer) num FROM days -- left join other tables here to get counts, I'm using random group by days. #sum_(k=1)^nb=nb# So. For example, the sum given by, means to sum an infinite number of terms as, The value of an infinite sum may be ∞ (in this case the sum is infinite). Definition: Arithmetic sequence is a list of numbers where each number is equal to the previous number, plus a constant. Using the while loop to calculate sum : While Loop « Statement Control « Java Tutorial. To add floating point values with extended precision, see math. Series calculator allows to calculate online the sum of the terms of the sequence whose index is between the lower and the upper bound. I can''t seem to find one of those. Build your own widget. A harmonic sequence is a sequence of numbers whose reciprocals form an arithmetic sequence. Here is the complete Java program with sample outputs. An Efficient solution to solve the sum of geometric series where first term is a and common ration is r is by the formula :-sum of series = a(1 – r n)/(1 – r). Let, t n be the n th term of AP, then (n+1) th term of can be calculated as (n+1) th = t n + D where D is the common difference (n+1) th - t n The formula to calculate N th term t n = a + (n – 1)d; where, a is first term of AP and d is the common difference. If Sum > Limit, terminate the repitition, otherwise. In the sequence a sub 1 (first term), a sub 2 (second term), a sub 3 (third term)…. Sum uses the standard Wolfram Language iteration specification. The last number n is input by the user. Σ is the symbol used to denote sum. Infinite Geometric Series Calculator is a free | {
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by the user. Σ is the symbol used to denote sum. Infinite Geometric Series Calculator is a free online tool that displays the sum of the infinite geometric sequence. Increment Number by one. Question: Find the sum of (1/2) +(1/6) + (1/12) + + (1/9900) without using a calculator. Sequence calculator allows to calculate online the terms of the sequence whose index is between two limits. 63333333 I'm having difficulty estimating how far this partial sum is away from the total sum. The blog talks about variety of topics on Embedded System, 8085 microprocessor, 8051 microcontroller, ARM Architecture, C2000 Architecture, C28x, AVR and many many more. Instructions: Use this step-by-step Geometric Series Calculator, to compute the sum of an infinite geometric series by providing the initial term $$a$$ and the constant ratio $$r$$. Arithmetic Sequence. After learning so much about development in Python, I thought this article would be interesting for readers and to myself… This is about 5 different ways of calculating Fibonacci numbers in Python [sourcecode language=”python”] ## Example 1: Using looping technique def fib(n): a,b = 1,1 for i in range(n-1): a,b = b,a+b return a print … Continue reading 5 Ways of Fibonacci in Python →. The 'nth' term is a formula with 'n' in it which enables you to find any term of a sequence without having to go up from one term to the next. 2, ALPHA ([I], 1, 1 0) EXE, and you should get an answer of 9. A sequence is a numbered list of values, produced by the calculation of a formula. C Program to calculate sum of 5 subjects and find percentage [crayon-5f51beda86fcb863628413/] Output : [crayon-5f51beda86fd2634443803/]. This is a free javascript calculator. The Fibonnacci numbers are also known as the Fibonacci series. Sequence Calculator, Sequence Examples. Improve your math knowledge with free questions in "Find the sum of an arithmetic series" and thousands of other math skills. Finding a general expression for a partial sum by induction and | {
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and thousands of other math skills. Finding a general expression for a partial sum by induction and then finding the limit of this partial sum is a perfectly valid technique. How to find the sum of a finite Arithmetic Series! s n = n(t 1 + t n)/2 To find the sum of a finite arithmetic series, you need to know three things. Without using loops, calculate the sum of the following series for the first n terms. You may want to review the basics of geometric sequences or finding formulas. Notice, that if x = 1 then, in the series, we are simply addi ng up an infinite number of 1's and of course the sum goes to infinity. A good sequence to start with is the Fibonacci sequence. Repeat the following: a. About this calculator. sum_arithmetic_series FIRST LAST SKIP NB. #sum_(k=1)^nk=(n(n+1))/2# and. Definition: Geometric sequence is a list of numbers where each term is obtained by multiplying the previous term by a constant. How do we apply these useful rules to this question? First, calculate the average of the first and. This formula shows that a constant factor in the summands can be taken out of the sum. I have made this small program to help me add numbers and find the sum, it runs but no output. 2) ∧ 2 × 0. LED series current limiting resistor calculator - useful when designing circuits with a single LED or series/parallel LED arrays - for both the common small-current (20mA) LEDs and the more expensive, high power LEDs with currents up to a few Amperes. #sum_(k=1)^n(ak+b)# where a and b are constants. Free Summation Calculator. If S n tends to a limit as n tends to infinity, the limit is called the sum to infinity of the series. To add floating point values with extended precision, see math. 0 / k if k % 2 else -1. Fibonacci sequence formula; Golden ratio convergence; Fibonacci sequence table; Fibonacci sequence calculator; C++ code of Fibonacci function; Fibonacci sequence formula. 78 degrees Fahrenheit. Click the link for more information. The n-th partial sum of a | {
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formula. 78 degrees Fahrenheit. Click the link for more information. The n-th partial sum of a series is the sum of the first n terms. asked by ryan on March 15, 2017; Algebra. White Rose Maths has prepared a series of Maths lessons online for Year 4, FREE videos and worksheets. This calculator computes n-th term and sum of geometric progression person_outline Timur schedule 2011-07-16 04:17:35 Geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. We start with the general formula for an arithmetic sequence of $$n$$ terms and sum it from the first term ($$a$$) to the last term in the sequence ($$l$$):. Explain why or why not. This is a very versatile calculator that will output sequences and allow you to calculate the sum of a sequence between a starting item and an n-th term, as well as tell you the value of the n-th term of interest. Calculate the rolling quantile. Since an arithmetic sequence always has an unbounded long-term behavior, we are always restricted to adding a finite number of terms. For this reason, Sum evaluates to null instead of to zero for an empty sequence or for a sequence that contains only nulls. So 355 minus 289. If the above series converges, then the remainder R N = S - S N (where S is the exact sum of the infinite series and S N is the sum of the first N terms of the series) is bounded by 0< = R N <= (N. Limits capture the long-term behavior of a sequence and are thus very useful in bounding them. The sum to infinity of a geometric progression. This version provided Java with a functional aspect by introducing concepts such as functional interfaces, lambda expressions, streams, etc. Sum of 32-bit integer quantities is not computed by using 64-bit results, and overflow can occur for the LINQ to SQL translation of Sum. Besides finding the sum of a number sequence online, server finds the partial sum of a series online. NPV = | {
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finding the sum of a number sequence online, server finds the partial sum of a series online. NPV = Sum CF* ((1+2%)/(1+D))^N When N is infinite, after simplification, NPV = CF * 1 / (1 - r) where r = (1+2%)/(1+D)--A+ V. Solving mathematical problems online for free. Since there's two equally likely options, you'd expect a run to last for two flips. See full list on gigacalculator. Sum of sequence calculator. And, thanks to the Internet, it's easier than ever to follow in their footsteps (or just finish your homework or study for that next big. Therefore, the equation could be 4n. A geometric sequence refers to a sequence wherein each of the numbers is the previous number multiplied by a constant value or the common ratio. Calculate the sum of special series or number of sequences. Sum of Arithmetic Sequence Formula. com Task : To find the sum of all the elements in a list. Leonhard Euler continued this study and in the process solved many. Note that a series is an indicated sum of the terms of a sequence!! In this section, we work only with finite series and the related sums. Such series appear in many areas of modern mathematics. In the previous section we started looking at writing down a power series representation of a function. Geometric Sequence. The free tool below will allow you to calculate the summation of an expression. To determine if the series is convergent we first need to get our hands on a formula for the general term in the sequence of partial sums. Guidelines to use the calculator If you select a n, n is the nth term of the sequence If you select S n, n is the first n term of the sequence For more information on how to find the common difference or sum, see this lesson Geometric sequence. skipna bool, default True. This free number sequence calculator can determine the terms (as well as the sum of all terms) of an arithmetic, geometric, or Fibonacci sequence. Question: Find the sum of (1/2) +(1/6) + (1/12) + + (1/9900) without using a calculator. | {
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sequence. Question: Find the sum of (1/2) +(1/6) + (1/12) + + (1/9900) without using a calculator. Store the result in an array. Perform addition to find the sum of two or more fractions. 8 Given two terms in an arithmetic sequence find the recursive formula. This calculator will find the infinite sum of arithmetic, geometric, power, and binomial series, as well as the partial sum, with steps shown (if possible). Hi everyone, the picture below is the sum of a series that have infinity limit, and i dont know the suitable function to deal with the problem. A sequence is a list of numbers, geometric shapes or other objects, that follow a specific pattern. For this reason, Sum evaluates to null instead of to zero for an empty sequence or for a sequence that contains only nulls. \$1000000007\$ is a prime number, so division by \$2^L\$ modulo 1000000007 is a multiplication by a multiplicative inverse of 2 modulo 1000000007 (which is trivial to find) to the same power. Calculate the sum of the series ∑ n = 1 ∞ a n whose partial sums are given. However, policy-makers disagree on the merits of this approach. Denote this partial sum by S n. An example of the sequence can be seen as follows:. Java program to calculate the sum of GP series. Fourth, we recall the sum of the X 2 and subtract 240. Work out: a. Online adding calculator. Sum of the series 1^1 + 2^2 + 3^3 + …. You may see the formula written as: Sum, S n, of n terms of an arithmetic series. The amount it increases or decreases by is known as the common difference. day ) select day, num, sum(num) over. +n and prints the result on the compiler screen. Two consecutive numbers in this series are in a ' Golden Ratio '. Flowchart. com allows you to find the sum of a series online. The Sum (Summation) Calculator is used to calculate the total summation of any set of numbers. 63333333 I'm having difficulty estimating how far this partial sum is away from the total sum. Such an argument was given by Nicolas Oresme (1323 - | {
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