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I am supposed to prove that $(1/1) + (1/3) + (1/6) + \dots + (1/t_n) < 2$.
The hint is that
$$
\frac2{n(n+1)} = 2\left(\frac1{n} - \frac1{n+1}\right)
$$
However, I was thinking that if you take $ \frac{1}{t_n} + \frac{1}{t_{n+1}} = \left(\frac1{t_{\frac{n}2}}\right)\left(\frac12\right)$ where $n$ is even.
... |
I have a probability space
$\omega = 2^{\{1,\ldots,n\}}$
$\sigma$-algebra $2^\omega$
and $P(\{s\})=(p^{|s|})*(1-p)^{(n-|s|)}$
I assume that $n=2k$,$k$ natural number
I need to find a random variable that will distribute like $\mathrm{Bin}(k,p^2)$
Can you help me with this please?
Thanks.
benny |
I am supposed to prove that $(1/1) + (1/3) + (1/6) + \dots + (1/t_n) < 2$.
The hint is that
$$
\frac2{n(n+1)} = 2\left(\frac1{n} - \frac1{n+1}\right)
$$
However, I was thinking that if you take $ 2\left(\frac{1}{t_n} + \frac{1}{t_{n+1}}\right) = \frac1{t_{\frac{n}2}}$ where $n$ is even.
So my logic is you ... |
How do I find generators of <a href="http://en.wikipedia.org/wiki/Octahedral_symmetry">Octahedral</a> group when viewing it as a subgroup of S8?
<b>Update</b>: Alex Bartel's suggestion gives the following permutations and Cayley graph
<img src="http://yaroslavvb.com/upload/save/math.se-cube.png">
<img src="http:... |
I have one incident edges and multiple outgoing Edges, for which I want to pick an outgoing edge such that the angles between the outgoing edge and the incoming edge is the smallest of all. We know the coordinates for the vertex $V$ .
The angle must start from the incoming edge ($e_1$) and ends at another edge ($e_2... |
I have one incident edges and multiple outgoing Edges, for which I want to pick an outgoing edge such that the angles between the outgoing edge and the incoming edge is the smallest of all. We know the coordinates for the vertex $V$ .
The angle must start from the incoming edge ($e_1$) and ends at another edge ($e_2... |
I have one incident edges and multiple outgoing Edges, for which I want to pick an outgoing edge such that the angles between the outgoing edge and the incoming edge is the smallest of all. We know the coordinates for the vertex $V$ .
The angle must start from the incoming edge ($e_1$) and ends at another edge ($e_2... |
The language described in (2) is not context free. Take the word $a^nba^nba^nb$ for $n$ large enough, and use the pumping lemma to get a word not in the language.
As for (1), consider a grammar in CNF (Chomsky Normal Form) generating that language. Let $n = |\Sigma|$. Consider all $S=(2n)!/2^n$ saturated words conta... |
I have one incident edges and multiple outgoing Edges, for which I want to pick an outgoing edge such that the angles between the outgoing edge and the incoming edge is the smallest of all. We know the coordinates for the vertex $V$ .
The angle must start from the incoming edge ($e_1$) and ends at another edge ($e_2... |
Here is the problem:
We have a square with some numbers from 1..N in some cells. It's needed to determine if it can be completed to a magic square.
Examples:
2 _ 6 2 7 6
_ 5 1 >>> 9 5 1
4 3 _ 4 3 8
7 _ _
9 _ _ >>> NO SOLUTION
8 _ _
Is this problem NP-complete... |
Is half-filled magic square problem NP-complete? |
The answer to your specific question is *yes*: if you are given a set $X$, then (even in practice!) you may choose any collection $\{U_i\}_{i \in I}$ of subsets of $X$ which (i) contains the empty set and $X$ itself, (ii) is closed under the operation of finite intersections and (iii) is closed under the operation of a... |
The answer to your specific question is *yes*: if you are given a set $X$, then (even in practice!) you may choose any collection $\{U_i\}_{i \in I}$ of subsets of $X$ which (i) contains the empty set and $X$ itself, (ii) is closed under the operation of finite intersections and (iii) is closed under the operation of a... |
Here is the problem:
We have a square with some numbers from 1..N in some cells. It's needed to determine if it can be completed to a magic square.
Examples:
2 _ 6 2 7 6
_ 5 1 >>> 9 5 1
4 3 _ 4 3 8
7 _ _
9 _ _ >>> NO SOLUTION
8 _ _
Is this problem NP-complete... |
I am looking for a generalized normal distribution with mean $\mu$ and variance $\sigma^2$ but with an additional parameter for the kurtosis to add fat-tails. I can't find it - can anybody help? |
Suppose I have an inscribed simplex which has n+1 vertices, and the diameter of the hypersphere is d. I have a point x inside this simplex, is it true that the distance between x and x's nearest vertex is less than d/2 ? |
Please help me in understanding the difference between this two question:
**How many ways can $4$ men and $4$ women stand in a line if the women and men alternate ?**
**How many ways can $4$ men and $4$ women stand in a line if no two women are together ?**
We employ a bit of different strategies for solving t... |
Suppose I have an inscribed simplex which has n+1 vertices, and the diameter of the hypersphere is d. I have a point x inside this simplex, is it true that the distance between x and x's nearest vertex is not larger than d/2 ? |
These are questions given for an assignment and I REALLY need to get a good grade so please could you guys check over them and give some pointers where I've gone wrong?
thanks.
$$\sum_0^\infty {\frac{(2x-8)^{n}}{4^{n}(n^{2}+2)}}$$
$$a_n = \frac{(2x-8)^{n}}{4^{n}(n^{2}+2)} , a_{n+1} =
\frac{(2x-8)^{n+1}}{4^{... |
Please help me in understanding the difference between this two question:
**How many ways can $4$ men and $4$ women stand in a line if the women and men alternate ?**
**How many ways can $4$ men and $4$ women stand in a line if no two women are together ?**
We employ a bit of different strategies for solving t... |
These are questions given for an assignment and I REALLY need to get a good grade so please could you guys check over them and give some pointers where I've gone wrong?
thanks.
$$\sum_0^\infty {\frac{(2x-8)^{n}}{4^{n}(n^{2}+2)}}$$
$$a_n = \frac{(2x-8)^{n}}{4^{n}(n^{2}+2)} , a_{n+1} =
\frac{(2x-8)^{n+1}}{4^{... |
In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?
I just dipped into a book, *The Drunkard's Walk - How Randomness Rules Our Lives*, by Leonard Mlodinow, Vintage Books, 2008. On p.107 Mlodinow says the chances are 1 in 3.
It seems obvious to me t... |
In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? |
we all know that 31,331,3331,33331,333331,3333331,33333331 all are primes. Here we prepend the digit 3 to 31, to get a list of 7 primes.This gives me the following thought:
Let $D = \{\text{all possible nonnull finite digit strings}\}$, $D' = \{\text{all things in D that do not start with "0"}\}$. Define a functio... |
Prepending strings and primes? |
These are questions given for an assignment and I REALLY need to get a good grade so please could you guys check over them and give some pointers where I've gone wrong?
thanks.
$$\sum_0^\infty {\frac{(2x-8)^{n}}{4^{n}(n^{2}+2)}}$$
\begin{align}
a_n &= \frac{(2x-8)^{n}}{4^{n}(n^{2}+2)}.\\\
a_{n+1} &=
\fra... |
I could use some help with proving the following irreducibility criterion. (It came up in class and got me interested.)
Let _p_ be a prime. For an integer $n = p^k n_0$, where _p_ doesn't divide $n_0$, set: $e_p(n) = k$. Let $f(x) = a_n x^n + \cdots + a_1 x + a_0$ be a polynomial with integer coefficients. If:
1.... |
Here's some idea.
Suppose you sample $4n$ families with two children, where $n$ is a very large integer. Then, with a very high probability, about $n$ of them have two boys, about $n$ have two girls, and about $2n$ have one girl and one boy.
Since you ignore the families with two boys, the desired probability is gi... |
How to prove that the group of order 28 which have normal subgroup of order 4 is Abelian group? |
Proof - Abelian group? |
It really depends on what knowledge you have to restrict the probability space. (The original question is a little ambiguous, but I agree that 1/3 is the best answer.)
The marble example and the example of the firstborn daughter are the same situation-- the second trial is independent, and the odds are 1/2. In the... |
In <a href="http://math.stackexchange.com/questions/12488/for-a-planar-graph-find-the-algorithm-that-constructs-a-cycle-basis-with-each-e"> your previous question</a> better use was made of graph theory terminology: you refer to the graph as planar and to a cycle basis.
Assuming you understand those terms, perhaps ... |
Hint: For this problem, you first have to select two of the symbols from your list. How many ways can that be done? Second, you have to select which $n_1$ positions in the word can take the first of the two symbols. How many ways can that be done? As the two selections are independent, you then multiply the two nu... |
Let $G$ be the group of order $28$, $N$ the normal subgroup of order $4$, and let $K = G/N$ be the quotient group, which has order $7$ and is therefore cyclic. By the [Schur-Zassenhaus Theorem][1], $G$ is a semidirect product of $N$ and $K$. It will be abelian iff the semidirect product is actually direct.
In or... |
Consider the case where we have a bag of 'N' unique marbles, and we randomly select all 'S' possible sets of size 'm' from the bag (with replacement) under the following constraints:
(1) - Any set of 'm' marbles must have all unique elements, i.e. there must be no duplicate copies of a marble in any given set.
(... |
Given a natural number a.Do we have a proof that there are infinitely many natural numbers not of the form anm+/-m+/n n, m natural? I give a proof that for a=6 the question is equivalent to the twin prime conjecture so it is known that we don't have any proof.But what about the other values of a?:
There are infinitely... |
do we have a proof of the infinitness? |
Suppose $z = a+bi$ and $w = u+iv$. Let $\displaystyle a = \left(\frac{|w|+u}{2} \right)^{1/2}$ and $\displaystyle b = \left(\frac{|w|-u}{2} \right)^{1/2}$. Show that $z^2 = w$ if $v \geq 0$ and $(\bar{z})^{2} = w$ if $v \leq 0$.
So $|w| = u^2+v^2$. So this is just a matter of computing $(a+bi)^2$ and $(a-bi)^2$ and ... |
In <a href="http://math.stackexchange.com/questions/12488/for-a-planar-graph-find-the-algorithm-that-constructs-a-cycle-basis-with-each-e"> your previous question</a> better use was made of graph theory terminology: you refer to the graph as planar and to a cycle basis.
Assuming you understand those terms, perhaps ... |
Here is an example. Suppose you have two implicit differentiable functions
$$F(x,y,z,u,v)=0,\qquad G(x,y,z,u,v)=0$$
and the functions, also differentiable, $u=f(x,y,z)$ and $v=g(x,y,z)$ such
that
$$F(x,y,z,f(x,y,z),g(x,y,z))=0,\qquad G(x,y,z,f(x,y,z),g(x,y,z))=0.$$
If you differentiate $F$ and $G$, you ... |
Suppose $z = a+bi$ and $w = u+iv$. Let $\displaystyle a = \left(\frac{|w|+u}{2} \right)^{1/2}$ and $\displaystyle b = \left(\frac{|w|-u}{2} \right)^{1/2}$. Show that $z^2 = w$ if $v \geq 0$ and $(\bar{z})^{2} = w$ if $v \leq 0$.
So $|w| = \sqrt{u^2+v^2}$. So this is just a matter of computing $(a+bi)^2$ and $(a-bi)^... |
Given a natural number a.Do we have a proof that there are infinitely many natural numbers not of the form $anm+/- m +/- n$ n, m natural? I give a proof that for a=6 the question is equivalent to the twin prime conjecture so it is known that we don't have any proof.But what about the other values of a?:
There are infi... |
I think it's an interesting coincidence that the [locker puzzle][1] and this [puzzle about duplicate array entries (see problem 6b)][2] have such similar solutions. _Spoiler alert!_ Don't read on if you want to solve these puzzles yourself first.
In both solutions we consider a collection of labeled boxes, each wit... |
What do lockers and duplicate array entries have in common? |
What's the reasoning behind assigning a proposition to a set, like $A=p(x)$ and $B=q(x)$? Is it because any set of elements has some property that defines all elements in it?
$$
\begin{align*}
A \cup (B \cap C) &= \{x|p(x) \vee (q(x) \wedge r(x))\}= \\\
&= \{x| (p(x)\vee q(x)) \wedge (p(x) \vee r(x)) \} \\\
&= \{x... |
What's the reasoning behind assigning a proposition to a set, like $A=p(x)$ and $B=q(x)$? |
I think it's an interesting coincidence that the [locker puzzle][1] and this [puzzle about duplicate array entries (see problem 6b)][2] have such similar solutions. _Spoiler alert!_ Don't read on if you want to solve these puzzles yourself first (they're two of the best puzzles I've ever seen).
In both solutions we... |
Suppose $z$ is a complex number. Prove that there exists an $r \geq 0$ and a complex number $w$ (with $|w| = 1$) such that $z = rw$. Does $z$ uniquely determine $r$ and $w$?
Let $z = a+bi$. Then $|z| = \sqrt{a^2+b^2}$. So take $r = |z|$. It seems like one can take $w$ to be any complex number such that $|w| = 1$... |
Just like the better-known [Monty Hall problem][1], this is a question about knowledge, and knowledge-of-knowledge.
In the Monty Hall problem, [the probability depends on whether or not Monty *knows* where the car is][2]. If Monty knows where the car is, he will *always* open a door with a goat, and switching will ... |
Just like the better-known [Monty Hall problem][1], this is a question about knowledge, and knowledge-of-knowledge.
In the Monty Hall problem, [the probability depends on whether or not Monty *knows* where the car is][2]. If Monty knows where the car is, he will *always* open a door with a goat, and switching will ... |
I think it's an interesting coincidence that the [locker puzzle][1] and this [puzzle about duplicate array entries (see problem 6b)][2] have such similar solutions. _Spoiler alert!_ Don't read on if you want to solve these puzzles yourself first (they're two of the best puzzles I've ever seen).
In both solutions we... |
Two seemingly unrelated puzzles have very similar solutions; what's the connection? |
I know that if $f$ is a Lebesgue measurable function on $[a,b]$ then there exists a continuous function $g$ such that $|f(x)-g(x)|< \epsilion$ for all $x\in [a,b]\P$ where the measure of $P$ is less than $\epsilion$.
This seems to imply that every Lebesgue measurable function on $\mathbb{R}$ is the pointwise limit... |
Is every Lebesgue measurable function on $\mathbb{R}$ the pointwise limit of continuous functions? |
[Crossposted on Mathoverflow.][1]
Given a natural number $a$, are there infinitely many natural numbers <i>not</i> of the form $anm \pm m \pm n$, $n, m$ natural?
I give a proof that for $a=6$ the question is equivalent to the twin prime conjecture so it is known that we don't have any proof. But what about other ... |
I know that if $f$ is a Lebesgue measurable function on $[a,b]$ then there exists a continuous function $g$ such that $|f(x)-g(x)|< \epsilon$ for all $x\in [a,b]\setminus P$ where the measure of $P$ is less than $\epsilon$.
This seems to imply that every Lebesgue measurable function on $\mathbb{R}$ is the pointwis... |
Let $L$ be the language of words in which each symbol appears an even number of times. We give an exponential lower bound (in $n = |\Sigma|$) on the size of a Chomsky Normal Form (CNF) grammar for $L$. This implies an exponential lower bound on the size of any context-free grammar for $L$, since (according to Wikipedia... |
When solving questions like these:
> Let $f(x)$ be a real function. Find
> $f(0.1)$ using its Taylor expansion
> such that the error is less than
> $10^{-3}$. Find the lowest degree of
> Taylor polynomial needed.
(if someone can rephrase the question so its more clear that would be great. I didn't quite nail ... |
Why is the remainder function $R_{n}(x)$ decreasing? |
I have an HP 50g graphing calculator and I am using it to calculate the standard deviation of some data. In the statistics calculation there is a type which can have two values:
Sample
Population
I didn't change it, but I kept getting the wrong results for the standard deviation. When I changed it to "Population... |
IF $A$,$B$,$C$ are finite sets then,
Number of elements in exactly one of sets $A$,$B$,$C$:$$n(A)+n(B)+n(C)-2 \times n(A \cap B)-2 \times n(A \cap C)-2 \times n(C \cap B) +n(A \cap B \cap C)$$
Number of elements in exactly two of the sets $A$,$B$,$C$: $$n(A \cap B)+(A \cap C)+n(C \cap B) - 3 \times n(A \cap B \ca... |
How to prove the following ? |
**What is the integral of a $k$-dimensional (multivariate) Gaussian over a convex $k$-polytope?**
Here I am specifically interested in $k\in\{2,3\}$, but insight on the general problem would also be appreciated. I am guessing that there is no closed form in the general case, but what if we restrict the number of ... |
How many Symmetric relations are there for an n-element set?
Thank you. |
How Many Symetric Relation on a Finite Set? |
I have an HP 50g graphing calculator and I am using it to calculate the standard deviation of some data. In the statistics calculation there is a type which can have two values:
Sample
Population
I didn't change it, but I kept getting the wrong results for the standard deviation. When I changed it to "Population... |
If $K_1$ and $K_2$ are subfields of a pre-chosen $\overline{\mathbb{Q}_p}$, and if they're both unramified at $p$, and $[K_1:\mathbb{Q}_p]=[K_2:\mathbb{Q}_p]$, does that imply that $K_1=K_2$?
My intuition says that this is true because all that's happening is that we're extending the residue field, and there's only ... |
Is an unramified cover of the p-adics determined by its degree? |
during a test i was given the following question:
What is 10+15+20+25+...+1490+1495+1500=?
After process of elimination and guessing, i came up with an answer. Can anyone tell me a simple way to calculate this problem without having to try to actually add the numbers? |
**Hint 1 -** Is there anything you can factor out of all the numbers?
**Hint 2 -** Do you know how to sum $1 + 2 + \cdots + n$ in a simple way? |
How many symmetric relations are there for an $n$-element set?
Thank you. |
How Many Symmetric Relations on a Finite Set? |
Taken from Rudin's *Principles of Mathematical Analysis*, 3rd edition (Chapter 1, Exercise 16):
Suppose $k\geq 3, x,y\in\mathbb{R}^k, |x-y|=d>0$, and $r>0$. Prove:
(a) If $2r > d$, there are infinitely many $z\in\mathbb{R}k$ such that $|z-x|=|z-y|=r$
(b) If $2r=d$ there is exactly one such $z$.
(c) If $2r <... |
Hint: For this problem, you first have to select two of the symbols from your list. How many ways can that be done? Should the order matter? Second, you have to select which $n_1$ positions in the word can take the first of the two symbols. How many ways can that be done? As the two selections are independent, you... |
Taken from Rudin's *Principles of Mathematical Analysis*, 3rd edition (Chapter 1, Exercise 16):
Suppose $k\geq 3, x,y\in\mathbb{R}^k, |x-y|=d>0$, and $r>0$. Prove:
(a) If $2r > d$, there are infinitely many $z\in\mathbb{R}^k$ such that $|z-x|=|z-y|=r$
(b) If $2r=d$ there is exactly one such $z$.
(c) If $2r ... |
Just like the better-known [Monty Hall problem][1], this is a question about knowledge, and knowledge-of-knowledge.
In the Monty Hall problem, [the probability depends on whether or not Monty *knows* where the car is][2]. If Monty knows where the car is, he will *always* open a door with a goat, and switching will ... |
I think it's an interesting coincidence that the [locker puzzle][1] and this [puzzle about duplicate array entries (see problem 6b)][2] have such similar solutions. _Spoiler alert!_ Don't read on if you want to solve these puzzles yourself first (they're two of the best puzzles I've ever seen).
In both solutions we... |
I think it's an interesting coincidence that the [locker puzzle][1] and this [puzzle about duplicate array entries (see problem 6b)][2] have such similar solutions. _Spoiler alert!_ Don't read on if you want to solve these puzzles yourself first (they're two of the best puzzles I've ever seen).
In both solutions we... |
**Hint 1** - Is there anything you can factor out of all the numbers?
**Hint 2** - Do you know how to sum $1 + 2 + \cdots + n$ in a simple way?
**EDIT** - Your process in the comments is not wrong, but I don't feel like it's very intuitive.
I do think it's immediately clear that all the numbers in your sum ... |
Let $X \geq 1$ be a integer r.v. with $E[X]=\mu$. Let $X_i$ be a sequence of iid rvs with the distribution of $X$. On the integer line, we start at $0$, and want to know the expected position after we first cross $K$, which is some fixed integer. Each next position is determined by adding $X_i$ to the previous position... |
I have a graph $G$ with a list of edges $E$, all the edges can form cycles, as shown below:
<img src="https://docs.google.com/drawings/pub?id=1Ux3b0FtuIMpVGX4e-AbblH05h2gIZII9u8pACRzymPE&w=960&h=720">
All the coordinates for the vertex $V$ **are known**.
Wat is the algorithm to find all the faces $F$? ... |
Let $X \geq 1$ be an integer r.v. with $E[X]=\mu$. Let $X_i$ be a sequence of iid rvs with the distribution of $X$. On the integer line, we start at $0$, and want to know the expected position after we first cross $K$, which is some fixed integer. Each next position is determined by adding $X_i$ to the previous positio... |
Let $G$ be a set of elements and $*$ a binary operation defined on the elements of $G$. Then $G$ is a group with respect to $*$ if the following holds:
- $a*(b*c) = (a*b)*c$ for all $a,b,c \in G$.
- There exists a left identity in $G$.
- For each $a \in G$, there exists a left inverse.
We could replace "left... |
Are there any good alogrithms which can be used to construct a prime greater than n, for arbitrary n?
There are some brute force approaches: for example, factoring (n+1)!. However, I'm looking for something significantly more efficient; preferably, polynomial in n.
Also, if anyone knows of a proof showing a bound... |
what is the process to solving this absolute value inequality? |
Are there any good algorithms which can be used to construct a prime greater than $n$, for arbitrary $n$?
There are some brute force approaches: for example, factoring $(n+1)!$. However, I'm looking for something significantly more efficient; preferably, polynomial in $n$.
Also, if anyone knows of a proof showing... |
I have a graph $G$ with a list of edges $E$, all the edges can form cycles, as shown below:
<img src="https://docs.google.com/drawings/pub?id=1Ux3b0FtuIMpVGX4e-AbblH05h2gIZII9u8pACRzymPE&w=960&h=720">
All the coordinates for the vertex $V$ **are known**.
Wat is the algorithm to find all the faces $F$? ... |
what is the process to solving this absolute value inequality: $4-|\frac{5y}{3} +4| \gt \frac{2}{5}$? |
what is the process to solving this absolute value inequality: $4-\left|\frac{5y}{3}+4\right| > \frac25$? |
What values of $Z_1$ and $Z_2$ make the five regions of the unit circle, shown below, equal in area? $\overline{Z_1}$ and $\overline{Z_2}$ are conjugates of $Z_1$ and $Z_2$; in other words they lie directly across the real axis from their counterparts.
![alt text][1]
[1]: https://i.stack.imgur.com/8cmP9.png |
I'm asking in the spirit of these [two][1] [questions][2]: can bitwise operations (AND, OR, XOR) be expressed in terms of other (more familiar?) functions?
I had been playing around with the bitwise operations in *Mathematica* a while back, and was at first struck by this identity for bitwise NOT: `~ n == -1 - n` (I... |
Differential equations of the form $M\,dx+N\,dy=0$ such that $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$ are said to be <i>exact</i>, because the left hand side of our equation is the <a href="http://en.wikipedia.org/wiki/Exact_differential">exact differential</a> of some function <i>f</i>, and the DE... |
Does the method for solving exact DEs generalize like this? |
In <a href="http://math.stackexchange.com/questions/12488/for-a-planar-graph-find-the-algorithm-that-constructs-a-cycle-basis-with-each-e"> your previous question</a> better use was made of graph theory terminology: you refer to the graph as planar and to a cycle basis.
Assuming you understand those terms, perhaps ... |
A general formula for this series? |
A general formula for $\sum (k-1)(k-2)(k-3)$? |
IF $A$,$B$,$C$ are finite sets then,
Number of elements in exactly one of sets $A$,$B$,$C$:$$n(A)+n(B)+n(C)-2 \times n(A \cap B)-2 \times n(A \cap C)-2 \times n(C \cap B) + 3 \times n(A \cap B \cap C)$$
Number of elements in exactly two of the sets $A$,$B$,$C$: $$n(A \cap B)+(A \cap C)+n(C \cap B) - 3 \times n(A ... |
These are questions given for an assignment and I REALLY need to get a good grade so please could you guys check over them and give some pointers where I've gone wrong?
thanks.
ed: I hope it's ok to overwrite the original post:
$$\sum_0^\infty {\frac{(2x-8)^{n}}{4^{n}(n^{2}+2)}}$$
$$a_n = \frac{(2x-8)^{... |
I want to show that $G$ is open in $X$ iff $\overline{G \cap \overline{A}} = \overline{G \cap A}$ for every $A \subset X$.
This direction $\rightarrow$ is easy.
I'm stuck in showing that $\overline{G \cap \overline{A}} = \overline{G \cap A}$ for every $A \subset X$ implies $G$ is open in $X$.
So I guess the g... |
Show that ${k+3 \choose 3}$ is the number of solutions to $x_1 + ... + x_4 = k$ in non-negative integers. Then $\sum_{k=0}^{n-4} {k+3 \choose 3}$ is the number of solutions to $x_1 + ... + x_5 = n-4$ in non-negative integers, which is...? |
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