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You can now traverse the sequence of equations once in either direction to show that $$\gcd(a,b) = \gcd(b,r_1) = \gcd(r_1,r_2) = \gcd(r_2,r_3) = ...\\ ... = \gcd(r_{n-2},r_{n-1}) = \gcd(r_{n-1},r_n) = \gcd(r_n,0) = r_n$$
Basically, instead of traversing the sequence of equations once in each direction, this proof buil... | {
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• You are using a proven result (that $\gcd$ is the biggest common divisor) to derive that $d\gt r_n$, which seems unfair to me. Please do not take this result for granted, which itself means that a proof derived using some other way was better to prove the topic of the post. – jiten Mar 29 '18 at 21:39
• what is your ... | {
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Prove/Disprove that if two sets have the same power set then they are the same set
I am really sure that if two sets have the same power set, then they are the same set. I just am wondering how does one exactly go about proving/showing this?
I'm usually wrong, so if anyone can show me an example where this fails, I'd... | {
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$A = B$ if and only if $\mathscr{P}(A) = \mathscr{P}(B)$.
-
+1 Is the statement that $\{ x \} \notin \mathscr P(B)$ evident or does it need proof? (It seems "clearly" true, but I do not know what to say if someone asks me to justify it.) – Srivatsan Sep 19 '11 at 2:44
I would just use the definition of subset. $D \sub... | {
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An alternative way to answer this old question: for all sets A and B,
$$\begin{array}{ll} & \mathcal{P}(A) = \mathcal{P}(B) \\ \equiv & \;\;\;\text{"extensionality"} \\ & \langle \forall V :: V \in \mathcal{P}(A) \equiv V \in \mathcal{P}(B) \rangle \\ \equiv & \;\;\;\text{"definition of \mathcal{P}, twice"} \\ & \lang... | {
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-
Instead of posting the same answer twice, you can post it once and point out that the question is the same. – Asaf Karagila Mar 16 '13 at 20:36
@AsafKaragila I'm sorry, I did not intend to post the same answer twice. Which two answers are you referring to? – Marnix Klooster Mar 16 '13 at 20:43
While not word for word... | {
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# Sphere equation given 4 points
Find the equation of the Sphere give the 4 points (3,2,1), (1,-2,-3), (2,1,3) and (-1,1,2).
The *failed* solution I tried is kinda straigh forward:
We need to find the center of the sphere.
Having the points:
$$p_{1}(3,2,1),\, p_{2}(1,-2,-3),\, p_{3}(2,1,3),\, p_{4}(-1,1,2)$$
2 Tr... | {
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I would cite the beautiful method from W.H.Beyer to find the center and radius of the sphere $(x-a)^2+(y-b^2)+(y-c)^2=R^2$
-
This is what I was refering to in my answer. – Claude Leibovici Aug 12 '14 at 9:53
Hi Claude ! Thanks for your notification. Of course, the method from W.H.Bayer in specific to the case of four... | {
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the center of the sphere is at $p=(24/19,-16/19,4/19)$
the squared radius is $r^2=4230/361$.
Here is my small GNU Maxima script
display2d : false;
/*
* purpose:
* given points x_k,
* calculate circumsphere center p and squared radius r^2
**/
my_A(d,k,x) := block(
[res,i,j],
res : zeromatrix(d,d),
for j : 0 t... | {
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The problem can be solved with this Maxima program:
/* define the four points p[1],...,p[4] */
p[1]:[3,2,1]; p[2]:[1,-2,-3]; p[3]:[2,1,3]; p[4]:[-1,1,2];
/* ceq is the equation of the circle
(u,v,w) is the center, r is the radius */
ceq:(x-u)^2+(y-v)^2+(z-w)^2=r^2;
/* plug in the points in the circle equation
to get... | {
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Question
# Let $$\alpha, \beta$$ be the roots of the equation $$ax^2+bx+c=0$$ and $$\alpha^4+\beta^4$$ be the roots of the equation $$px^2+qx+r=0$$, then the roots of the equation $$a^2px^2-4acpx+2c^2p+a^2q=0$$ are:
A
Always +ve
B
Always complex
C
Opposite in sign
D
Negative
Solution
## The correct option is C Oppo... | {
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Showing that $\lim_{x \to 1} \left(\frac{23}{1-x^{23}}-\frac{11}{1-x^{11}} \right)=6$
How does one evaluate the following limit? $$\lim_{x \to 1} \left(\frac{23}{1-x^{23}}-\frac{11}{1-x^{11}} \right)$$ The answer is $6$.
How does one justify this answer?
Edit: So it really was just combine the fraction and use L'hop... | {
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• I didn't expect this many responses to my question, but your approach is the clearest and most standard of all of the other approaches. Also, I didn't think combining the fraction was necessary but it is. Because I see that it is much better to deal with indeterminate forms of $\frac 00$ instead of $\infty - \infty$,... | {
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Using the identity $1-x^{2n+1} = (1-x)\sum_{k=0}^{2n} x^k$, we can rewrite \begin{align*} \frac{23}{1-x^{23}} - \frac{11}{1-x^{11}} &= \frac{1}{1-x}\left(\frac{23}{\sum_{k=0}^{22}x^k} - \frac{11}{\sum_{k=0}^{10}x^k} \right)\\ &= \frac{1}{1-x}\left(\frac{23\sum_{k=0}^{10}x^k}{\sum_{k=0}^{22}x^k\sum_{k=0}^{10}x^k} - \fra... | {
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• Darn, took much longer to type than expected. Oh, well. – Clement C. Sep 28 '16 at 20:55
• I tried an approach like yours originally but I got stuck at dealing with that $\frac 1{1-x}$. Your solution to this dilemma was simply to write $x=1+h$ (for $h > 0$)? I guess since we're taking the limit to $1$ from both sides... | {
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$$-\frac{1}{t}\left(1-11t-1+5t+O(t^2)\right)=6+O(t)$$
And your limit is $6$.
That is the same as
$$\lim_{x\to 0}\left[\frac{23}{1-(1-x)^{23}}-\frac{11}{1-(1-x)^{11}}\right]=\lim_{x\to 0}\left[\frac{23}{23x-253x^2}-\frac{11}{11-55x^2}\right]$$ (we exploited the binomial theorem and neglected terms with high order, si... | {
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• I don't like very much to "neglect terms". There is the notation $O(x^3)$ for this purpose (or $o(x^2)$ if you prefer). More correct, and less dangerous: what happens when all previous terms cancel, you are left with $0$ while there is indeed some nonzero function of $x$? For instance, how do you know in advance that... | {
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• How did you justify your second equality? Because $\frac 1{1-x} = \sum_{k=0}^\infty x^k$, an infinite series. How did you get finite series in the numerator? By squaring the denominator? – user373314 Sep 29 '16 at 8:07
• @user373314 Note that $1 - x^{n} = \left(1 - x\right)\left(1 + x + x^{2} + \cdots + x^{n - 1}\rig... | {
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$11 \sum_\limits{i=1}^{11} i + 12 \sum_\limits{i=1}^{11} i = (23)(11)(12)/2$
and the ratio $= 6$
Write $P_n(x)=1+x+x^2+\dots+x^n$; then our function is $$\frac{23P_{10}(x)-11P_{22}(x)}{(1-x)P_{22}(x)P_{10}(x)}$$ We can notice that $$\lim_{x\to1}P_{22}(x)P_{10}(x)=23\cdot11$$ so we just need to compute $$\lim_{x\to1}\... | {
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The following standard formula is well known $$\lim_{x \to 1}\frac{x^{n} - 1}{x - 1} = n = \lim_{t \to 0}\frac{(1 + t)^{n} - 1}{t}\tag{1}$$ and it appears that we can go very easily to the next step if $n$ is a positive integer and derive the formula $$\lim_{x \to 1}\frac{x^{n} - 1 - n(x - 1)}{(x - 1)^{2}} = \lim_{t \t... | {
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# Complex Line Integrals
This example shows how to calculate complex line integrals using the `'Waypoints'` option of the `integral` function. In MATLAB®, you use the `'Waypoints'` option to define a sequence of straight line paths from the first limit of integration to the first waypoint, from the first waypoint to t... | {
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### Integrate Along a Contour That Encloses No Poles
If any limit of integration or element of the waypoints vector is complex, then `integral` performs the integration over a sequence of straight line paths in the complex plane. The natural direction around a contour is counterclockwise; specifying a clockwise contou... | {
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# Proof that $A \cap B$ and $A \setminus B$ are disjoint.
I am trying to prove that $A \cap B$ and $A \setminus B$ are disjoint. Here is what I've done so far.
Is there anything that's wrong in my proof, and is there anything that can make it better?
Proof: $A \cap B$ and $A \setminus B$ are disjoint if $(A \cap B) ... | {
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The proof is correct, but it is a tad verbose.
If you are going to write a proof by contradiction, I recommend you say so up front.
You just need to show that $x$ cannot be in both $B$ and $B'.$
Alternatively, show $(A \cap B) \cap (A \cap B') = \emptyset$
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# Construct the field of complex numbers as the quotient ring of real polynomials
I am trying to construct the field of complex numbers as the quotient ring of real polynomials.
Suppose that
1. $$\mathbb C, \mathbb R$$ are the fields of complex and real numbers respectively.
2. $$\mathbb R [X]$$ is the ring of poly... | {
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1. Uniqueness
Suppose that $$s’$$ and $$r’$$ are other polynomials such that $$p=s’ q+r’$$ and $$\operatorname{deg} (r’) < \operatorname{deg}(q)$$. Then $$(s’-s) q=r-r’$$. If $$s’-s \neq 0$$ then, from $$\operatorname{deg} (p q) = \operatorname{deg}(p) + \operatorname{deg}(q)$$, we would get
$$\operatorname{deg} (r-r... | {
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• Thank you so much for your verification ;) Jun 29 '19 at 9:15
• I'm glad I could help. Jun 29 '19 at 9:35
Well, the quotient ring $$D = {\Bbb R}[x]/\langle x^2+1\rangle$$ is a field since $$x^2+1$$ is irreducible over $$\Bbb R$$.
Moreover, the quotient ring $$D$$ contains a zero of $$x^2+1$$, namely $$\bar x = x+\l... | {
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# In how many different ways can I sort balls of two different colors
Let's say, I have 4 yellow and 5 blue balls. How do I calculate in how many different orders I can place them? And what if I also have 3 red balls?
• maybe a title more clear could be "In how many different ways can I sort balls of two different co... | {
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For some reason I find it easier to think in terms of letters of a word being rearranged, and your problem is equivalent to asking how many permutations there are of the word YYYYBBBBB.
The formula for counting permutations of words with repeated letters (whose reasoning has been described by Noldorin) gives us the co... | {
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# Arbitrary vs. Random
I'm currently assisting a basic course where the students have to write some proofs. Most of them use terminology like "Let $x$ be a random integer", instead of "Let $x$ be an arbitrary integer" or plainly "Let $x$ be an integer". Am I doing the right thing in correcting them?
My point of view ... | {
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I originally wrote a more ambivalent response, but thinking about it further I've changed my mind.
It's clear that the phrase "let $x$ be a random integer" is mathematically . . . bad. What is at question is whether:
• it is misleading to the student,
• it is worth correcting,
• and as a bonus, whether it is worth ... | {
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Next, it would probably help the aforementioned fledglings if they were shown why the distinction is useful. One practical reason is simplicity. If one deals with an arbitrary integer $x$, all that is assumed is that $x \in \mathbb{Z}$. Could $x = 25$ be true? Of course! Could $x = 25$ be false? Certainly!
If, however... | {
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Actual dictionary definitions: Doing some quick dictionary searching for "arbitrary" gives the definition: "based on random choice or personal whim, rather than any reason or system." The definition given for "random" is "made, done, happening, or chosen without method or conscious decision." In fact, "random" is liste... | {
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So, inasmuch as it is a real process of coming up with an actual example of an arbitrary number, it might actually be a type of random number in the probabilistic modeling sense. Of course, in the actual mathematical context where the number is to be used, it is just an arbitrary number, e.g. to be plugged into an equa... | {
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# Why can't the ratio test be used for geometric series?
The ratio test says that, for $a_k\neq 0$, if $$\lim_{k\to\infty}\left|\frac{a_{k+1}}{a_k}\right|=L$$ exists, then if $0\leq L <1$, then $\sum_k a_k$ converges. If $L>1$, it diverges.
The notes I'm reading say that it's inadmissible to use the ratio test to tes... | {
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The usual proof of the ratio test is to compare the series to a geometric series. If $$\lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| = \alpha < 1,$$ then we have $$|a_{n+1}| < |a_n| \alpha$$ for all sufficiently large $n$. It then follows from an induction argument that $$|a_{n+k}| < |a_n| \alpha^k$$ for $n$ suf... | {
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While one could use the ratio test to establish the convergence of a geometric series (there is nothing stopping us!), it is typically poor style to rely on circular arguments as it can (potentially) lead to overlooking important hypotheses or exceptional cases. This is particularly important in a pedagogical setting, ... | {
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However, if you happen to be in the process of validating the ratio test, it would not be valid to use the ratio test to justify any of the facts you need — such as when geometric series converge — in its validation. You would have to first derive the facts about geometric series in some other fashion.
As for your not... | {
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• Here are the notes (look at the second bullet point on page 46) courses.maths.ox.ac.uk/node/view_material/1087 Sep 7, 2017 at 7:57
• and then point B on the problem sheet (page 2): "Why is it not admissible to establish the convergence or divergence of a geometric series by using the Ratio Test?" courses.maths.ox.ac.... | {
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Once we have a valid proof of the ratio test, then it can be applied to any series if it satisfies the hypotheses, including geometric series. It does not matter how the ratio test was proven. This is in no way circular reasoning even if the proof used results about geometric series. The idea of circular reasoning is a... | {
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• It's correct but still circular, like computing $\lim_{x \to 0} \frac{e^x-1}{x}$ using L'Hospital's rule. Ultimately if you expanded out the proof of the ratio test, you would find that in the body of that proof is the result you already want to prove being asserted without proof...unless you have some other proof of... | {
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Probability of getting heads in a coin toss
A fair coin is tossed five times. What is the probability of getting a sequence of three heads?
What i tried
Since the probability of getting a head and not getting a head is $0.5$. The probability of getting three heads is $0.5^3$ while there is a $0.5^2$ chance of not ge... | {
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The only allowable sequences are HHHTT, THHHT, TTHHH, HHHHT, THHHH, HTHHH, HHHTH, HHHHH.
• So the algorithm should be $(0.5)^3=1/8$? – ys wong Aug 24 '15 at 16:41
• No. How did you get that expression? To solve it: what is the probability of HHHTT? Of THHHT? Of HHHHH? – Patrick Stevens Aug 24 '15 at 16:43
• Okay Thank... | {
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This is just me, but I'd take a direct approach and (since you only have 5 events to consider) draw out a table or chart of all the possible outcomes for first only 2 or 3 events, see if you can scale up, then check your calculation against that.
I find when people aren't inwardly certain in a solid way about the algo... | {
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# Inconsistency in Histogram's “Probability” Binsize
Context
Let me define a Probability distribution (following the documentation and with some connection to this question)
DD = ProbabilityDistribution[(Sqrt[2]/\[Pi]) (1/(1 + x^4)), {x, -\[Infinity], \[Infinity]}];
which is normalized properly CDF[DD,1000]//N (*... | {
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But note the 1/4 factor in the PDF (corresponding to the binsize).
-
If you use "PDF" instead of "Probability", then the scaling comes out right... – Ｊ. Ｍ. Nov 2 '12 at 9:53
@J.M. so it was a trap! You knew the answer: how wicked! Well you can write it as a one-liner now :-) – chris Nov 2 '12 at 9:56
Well, you got a... | {
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# Prove that the following argument is valid
I'm asked to show that the following argument is valid:
P1) $$[E \lor (L \lor M)] \land (E \leftrightarrow F)$$
P2) $$L \rightarrow D$$
P3) $$D \rightarrow \neg L$$
C) $$E \lor M$$
Here is my work (so far):
P2) $$L \rightarrow D$$
1. $$\neg(\neg L) \rightarrow D$$ Pr... | {
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8) $E\vee M$ by disjunctive syllogism and 7,3.
Conclude that the argument is valid.
• thank you so much , but can you recommended me with a good book explain logic May 21, 2015 at 23:00
• It depends on the level at which you'd like to study logic, the topics you want to study, and the ways you'd eventually like to us... | {
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Here is the proof:
Here is a summary of the proof:
• Use conjunction elimination (∧E) to get the first conjunction from premise 1. We will not need the second conjunct from that premise.
• Since this conjunct is a disjunction consider both cases of the disjunction separately. If we can reach the desired conclusion in... | {
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# cube roots calculator | {
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To use the calcualor simply type any positive or negative number into the text box and hit the 'calculate' button. Square Root Calculator Geek was created by a group of math proffesionals with a strong desire to create simple, quick solutions for teachers and students. There is a process that appears a bit laborious at... | {
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or negative numbers (including decimals). For Polynomials of degree less than or equal to 4, the exact value of any roots (zeros) of the polynomial are returned. The Cube Root Calculator will calculate the cube root of any number and all you have to do to calculate the cube root is to just enter in any number and press... | {
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and content are provided "as is", free of charge, and without any warranty or guarantee. Wikipedia – Cube root, nth root, and Cube (algebra) detail the calculations involved with cube roots. The cube root calculator below will reduce any cube root to its simplest radical form as well as provide a brute force rounded ap... | {
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cube root calculator might come in handy whenever you need to calculate the cube root of any given positive or negative numbers (including decimals). It is called a "cube" root since multiplying a number by itself twice is how one finds the volume of a cube. Plotting the results from the cube root function, as calculat... | {
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this online calculator resource and information as provided on the page, you can use the following citation: Georgiev G.Z., "Cube Root Calculator", [online] Available at: https://www.gigacalculator.com/calculators/cube-root-calculator.php URL [Accessed Date: 27 Nov, 2020]. The cube roots from 1 to 10 for positive and n... | {
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Does Child Support Start From Date Of Separation, Citrus Rootstock For Sale, Chair Back Angle, Bible Characters Who Showed Love, Modern House Logo, Fallout 4 Weapon Mods Reddit, Ellie Animal Crossing House Exterior, Dean's Guacamole Dip Recall, Lenovo Ideapad Flex 5 14are05 Specs, Zucchini Pasta With Tomato Sauce Calor... | {
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It is currently 23 Jan 2018, 05:30
### GMAT Club Daily Prep
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Please give kudos, if u liked my post!
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When the going gets tough, the tough gets going...
The length of a rectangular floor is 16 feet and its width is 12 feet. [#permalink] 20 Sep 2017, 10:49
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# The length of a rectangul... | {
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# Difference between revisions of "2016 AMC 8 Problems/Problem 23"
## Problem 23
Two congruent circles centered at points $A$ and $B$ each pass through the other circle's center. The line containing both $A$ and $B$ is extended to intersect the circles at points $C$ and $D$. The circles intersect at two points, one o... | {
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## Video Solution
https://youtu.be/WJ0Hodj0h2o - Happytwin | {
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Fourier Series Of Sine Wave Fourier series usually include sine and cosine functions and can represent periodic functions in time or space or both. On this page, the Fourier Transforms for the sinusois sine and cosine function are determined. Fourier Series--Square Wave. The correct harmonic series is the one where the... | {
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(possibly infinite) sine wave to represent a periodicsignal. waveform in Table 15. We'll look at the cosine with frequency f=A cycles/second. Fourier Series Analysis { Fourier Series Analysis (C) 2005-2018 John F. Fourier series In the following chapters, we will look at methods for solving the PDEs described in Chapte... | {
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and plot it. FOURIER SERIES 7 On the other hand, the sum of the squares of the coe cients is: X1 n=1 2 n 2 = X1 n=1 4 n2: The formula is therefore telling us that X1 n=1 4 n2 = 2ˇ2 3 X1 n=1 1 n2 = ˇ2 6 This remarkable identity is actually correct, and was rst worked out by Euler. The function sin (x/2) twice as slow as... | {
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out. The Fourier coefficients , a k and b k , express the real and imaginary parts respectively of the spectrum while the coefficients c k of the complex Fourier series express the spectrum as a magnitude and phase. According to the Fourier theorem, a steady-state wave is composed of a series of sinusoidal components w... | {
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to the study of physics. Math 331, Fall 2017, Lecture 2, (c) Victor Matveev. (2) as, S(t) 2To -To 0 To 2Tot ; Question: Fig. As signal of n data points thus there are n/2 sinusoidal curves in a Fourier series. Graphing a Fourier Series. Fourier series approximations to a square wave The square wave is the 2 p-periodic ... | {
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function, so bm = 0 Ao = 1/pi∫sinωt from 0 to pi = 1/pi(-cos(ωt))/ω) from 0 to pi = 2/piω. Fourier series is a mathematical function that is formed by the sum of scaled sine and cosine functions, over an interval. If we suppose that any piecewise-continuous function can be represented by a Superposition of sines and co... | {
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with sine waves (known for centuries) and Stokes waves (known since the Stokes paper in 1847). Title and author: Fourier Series with Sound. Define sK(t) to be the signal containing K+1 Fourier terms. THE CONNECTION TO THE FOURIER SERIES IAN ALEVY Abstract. A1 and A2 respectively. Fourier Series on a bar of length l: Le... | {
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wave. The Fourier Transform for this type of signal is simply called the Fourier Transform. The time domain signal used in the Fourier series is periodic and continuous. Move the mouse over the white circles to see each term's contribution, in yellow. It was Fourier's discovery that any continuous repetitive wave could... | {
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& t=findgen(N) IDL> f=10*sin(2*!pi*t/32) + 20*randomn(seed,N) IDL> plot,f You can see that it is difficult to distinguish the sine wave from the noise. The width in the peak of the Fourier transform is a way of saying there is an uncertainty in the "true" value of the frequency. Within one period, the function is f(x) ... | {
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the exponential series for the following rectangular wave, given by. The graph of the function t 010; 2 f(x) = 8 <: x; 0 x 1 x+1; 1 0 other coefficients the even symmetry of the function is exploited to give. The Fourier Series is a shorthand mathematical description of a waveform. A pure sine wave can be converted int... | {
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zero is an excellent time to enter or exit a swing trade. 000000000000000 -4. Remark: If f is continuous on [0;1], then these two series also converge to f(x) at x= 0;1. It turns out that we have just the odd frequencies 1, 3, 5 in the square wave and they're multiplied by 4 over pi and they're divided by the frequency... | {
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it can be used to represent any function over the chosen interval. bk = { 4 πk if k is odd 0 if k is even. The Plancherel identity suggests that the Fourier transform is a one-to-one norm preserving map of the Hilbert space L2[1 ;1] onto itself (or to another copy of it-self). Are you trying to find the Fourier series ... | {
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does not involve the terms with sines and has the form: ${f\left( x \right) = \frac{{{a_0}}}{2} }+{ \sum\limits_{n = 1}^\infty {{a_n}\cos nx} ,}$ where the Fourier coefficients are given by the formulas \. Fourier Series - Sine Wave Synthesis. The application of Fourier-series method includes signal generators, power s... | {
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coefficients fb ng1 n=1 is the expression F(x) = X1 n=1 b nsin nˇx T Theorem. Spectrum from Fourier Series Plot a for Full-Wave Rectified Sinusoid F 0 1 / T 0 d Z 0 2SF 0 ( 4 1) 2 2 k a k S a k. Here, a sine function is full-wave rectified, meaning that the wave becomes positive wherever it would be negative. We will us... | {
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of distortion, known as slew rate distortion, which will not affect a pure sine wave. Fourier Series Grapher. 3 Fourier Cosine and Sine Series. Solution The simplest way is to start with the sine series for the square wave: SW(x)= 4 π sinx 1 + sin3x 3 + sin5x 5 + sin7x 7 +···. See full list on en. The Fourier coefficie... | {
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The computation and study of Fourier series is known as harmonic. The cosine form is also called the Harmonic form Fourier series or Polar form Fourier series. Depending on which boundary conditions apply, either the position or the lateral velocity of the string is modelled by a Fourier series. It’s a baffling concept... | {
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m m Again, we really need two such plots, one for the cosine series and another for the sine series. 1:10; y = sin (t); plot (t,y); Next add the third harmonic to the fundamental, and plot it. There are JavaScript (React) version, and Unity version. such waveform can be represented in series form based on the original ... | {
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of a Fourier expansion by building on the ideas of phasors, partials, and sinusoidal components that we introduced in the previous section. Plot this fundamental frequency. The graph of the function t 00 be a xed number and f(x) be a periodic function with period 2p, de ned on ( p;p). Either way the maximum slope of th... | {
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of sine wave curves: Above is Python (Blender) version. Complex Fourier Series The complex Fourier series is presented first with pe-riod 2π, then with general period. In fact, as we add terms in the Fourier series representa-. The primary reason that we use Fourier series is that we can better. Fourier Series of Even a... | {
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representing a function as a sum of trigonometric functions greatly simplifies the study of heat. • Consider, for example, a triangular waveform. The Fourier-space (i. Fourier series is just a means to represent a periodic signal as an infinite sum of sine wave components. Using the Fourier expansion, the frozen surfac... | {
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infinite series, where the individual elements of the series are expressed trigonometrically. ( n 2 π x / λ − n 2 π f 1 t). The waveforms in these figures were generated using truncated, finite-term version(s) of the Fourier series expansion for this waveform: The first figure shows the bipolar triangle wave (labelled ... | {
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plots the trig. or spectral phase of the Fourier series. Fourier Series of Even and Odd Functions. Furthermore, if the function of choice is periodic, then the Fourier series can be used to. A rectified sine wave is a periodic signal with a period equal to half of the full sinusoid, I would write the sine in exponentia... | {
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1 / T 0 d Z 0 2SF 0 ( 4 1) 2 2 k a k S a k. The cosine form is also called the Harmonic form Fourier series or Polar form Fourier series. Fourier series of a constant function f(x)=1 converges to an odd periodic extension of this function, which is a square wave. Fourier Analysis for Periodic Functions. The values of a... | {
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ng1 n=1 in a Fourier sine series F(x) are determined by. BACKGROUND IN FOURIER SERIES Jean Baptise Joseph Fourier (1768-1830) was the inventor of Fourier series in the late 1700s. Expression (1. (ii) The Fourier series of an odd function on the interval (p, p) is the sine series (4) where (5) EXAMPLE 1 Expansion in a S... | {
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2b:. For example, later we will Example of a Fourier Series - Square Wave Sketch the function for 3 cycles: f(t) = f(t + 8). Find the Fourier series of the resulting periodic function: w w w p L L E t t L L t u t, 2, 2 sin 0 0 0. Fourier series make use of the orthogonality relationships of the sine and cosine function... | {
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below. You can see that after rectification, the fundamental frequency is eliminated, and all the even harmonics are present. The next animation shown how the first few terms in the Fourier series approximates the periodic square wave function. Guitars and pianos operate on two different solutions of the wave equation.... | {
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corresponding to its frequency. Once you have a decent sine wave generator, the natural next step is to add a bunch together as a Fourier series and approximate common waveforms like the triangle wave or square wave. Build and use a Fourier Series analyzer. We thereby multiply our signal (target function) with an analy... | {
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3 on the page Definition of Fourier Series and Typical Examples, we can write the right side of the equation as the series \[{3x }={ \frac{6}{\pi }\sum\limits_{n = 1}^\infty {\frac{{{{\left( { - 1} \right)}^{n + 1}}}}{n}\sin n\pi x}. With this. The Fourier series of the above sawtooth wave is. The next animation shown ... | {
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function can be rewritten, thanks to Euler, using the identity:. Fourier series for a function. We can use Fourier Series to investigate. I've tried to learn about Fourier synthesis from many sources, but they all talk about the Fourier series instead of the Fourier transform, and they all say that for a pure wave all ... | {
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is a description of a waveform such as a square or triangle wave. If the wave shape is periodic, the frequencies of the partials are multiples of the fundamental frequency and are called the “harmonics” of the tone being played. • Fourier Cosine Series This is a half-range series consisting solely of cosines. Since a s... | {
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common waveforms as well. Spectrum from Fourier Series Plot a for Full-Wave Rectified Sinusoid F 0 1 / T 0 d Z 0 2SF 0 ( 4 1) 2 2 k a k S a k. 1, and take the sine of all the points. , slow) as ΠT(t). Let the integer m become a real number and let the coefficients, F m, become a function F(m). This demonstration shows ... | {
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signal (period. This version of the Fourier transform is called the Fourier Series. Here A1=A2, so the average is zero. Lec1: Fourier Series Associated Prof Dr. The Fourier series expansion for a square-wave is made up of a sum of odd harmonics, as shown here using MATLAB®. • Fourier Series: Represent any periodic func... | {
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# Evaluation of $\int_{0}^\infty \frac{\sin(x)}{x}e^{- x²} dx$
I have tried to evaluate the integral $$\int_{0}^\infty \frac{\sin(x)}{x}e^{- x^2} dx.$$ I used integration by parts but I did not succeed. Wolfram Alpha says that is convergent and it is equal to: $\frac \pi 2 \text{erf}({\frac 12})$ .
Is there any simpl... | {
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Also, differentiating (1) again with respect to the parameter $a$ yields $$I''(a) = -\int_0^\infty x e^{-x^2} \sin (ax) \, dx = -\frac{a}{2} I'(a).$$ If we set $u(a) = I'(a)$ the above second-order differential equation can be reduced to the following first-order differential equation $$u'(a) = -\frac{a}{2} u(a).$$ Sol... | {
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Following Sangchul Lee's hint. \begin{align} I:=\int^\infty_0 \frac{\sin(x)}{x}e^{-x^2}\,dx=\frac{1}{2}\int^\infty_{-\infty}\frac{\sin(x)}{x}e^{-x^2}\,dx=\frac{1}{4}\int^\infty_{-\infty}\int^1_{-1}e^{ixt}e^{-x^2}\,dt\,dx \end{align} Since $|e^{ixt}e^{-x^2}|=e^{-x^2}$ the double integral is clearly absolutely convergent... | {
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# How to evaluate a limit of the indeterminate form $(0/0)^0$
How to find the $\lim_{n \to \infty} \left(\dfrac{(n+1)(n+2)\cdots(n+2n)}{n^{2n}}\right)^{1/n}$? I know how to find it for the indeterminate form of $1^{\infty}$ by converting it into $0/0$ form, but this cannot be converted into any known indeterminate for... | {
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We have$^{(\dagger)}$ $$\frac{1}{2n}\sum_{k=1}^{2n}f\!\left(\frac{k}{2n}\right)\xrightarrow[n\to\infty]{} \int_0^1 f = \frac{1}{2}(3\ln 3 -2)$$ and by continuity of the exponential your limit will be $$\left(\frac{\prod_{k=1}^{2n}(n+k)}{n^{2n}}\right)^{\frac{1}{n}} \xrightarrow[n\to\infty]{} e^{2\int_0^1 f} = e^{3\ln 3... | {
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"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130567217291,
"lm_q1q2_score": 0.8425924411396762,
"lm_q2_score": 0.8519527982093668,
"openwebmath_perplexity": 610.2020085128898,
"openwebmath_score": 0.8673697113990784,
"tag... |
• How $\frac{n+k}{n^{2n}}$ becomes $\frac{n+k}{n}$? – Von Neumann Sep 9 '16 at 18:23
• @FourierTransform It did not. $\frac{1}{n^{2n}} \prod_{k=1}^{2n} (n+k)$ became $\prod_{k=1}^{2n} \frac{n+k}{n}$. – Clement C. Sep 9 '16 at 18:24
• Oh! Sorry. I read wrong. Great! – Von Neumann Sep 9 '16 at 18:25
• I know this comment... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130567217291,
"lm_q1q2_score": 0.8425924411396762,
"lm_q2_score": 0.8519527982093668,
"openwebmath_perplexity": 610.2020085128898,
"openwebmath_score": 0.8673697113990784,
"tag... |
$$\implies z = n(3\log_e 3 - 2)$$
$$\text{substituting the value of z in y}$$
$$y= e^{{1\over n}z}$$
$$\implies y= e^{3\log_e 3 - 2}$$
$$\text{Finally finding the limit}$$ $$\implies \lim_{n \to \infty} y = \lim_{n \to \infty} e^{3\log_e 3 - 2} = e^{3\log_e 3 - 2} = {e^{3\log_e 3}\over e^2} = {27\over e^2}$$
$$\co... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130567217291,
"lm_q1q2_score": 0.8425924411396762,
"lm_q2_score": 0.8519527982093668,
"openwebmath_perplexity": 610.2020085128898,
"openwebmath_score": 0.8673697113990784,
"tag... |
### RPN Calculations and the Stack
The central component of an RPN calculator is the stack. A calculator stack is like a stack of dishes. New dishes (numbers) are added at the top of the stack, and numbers are normally only removed from the top of the stack.
In an operation like 2+3, the 2 and 3 are called the operan... | {
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"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9416541561135441,
"lm_q1q2_score": 0.8425822215835271,
"lm_q2_score": 0.8947894682067639,
"openwebmath_perplexity": 1625.8453885811211,
"openwebmath_score": 0.6622502207756042,
"tags": null,
... |
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