5
votes
3answers
657 views

Preventing “proof by homework”?

I am doing problem 3d in the Prologue of Spivak: Prove $(a+b)^n = a^n + {n\choose1}a^{n-1}b + {n\choose2}a^{n-2}b^2 + ... + {n\choose n-1}ab^{n-1} + b^n$ I feel like my proof could pass off as ...
2
votes
1answer
74 views

Prove that $\displaystyle\sum_{j=m}^n\sum_{k=0}^{2m}{4j\choose 2k}{2j-k\choose 2m-k}={2n+2m+1\choose 4m+1}2^{4m-1}$

Let $n,m$ are positive integers satisfy the condition $n\ge m>0$ Prove that $\displaystyle\sum_{j=m}^n\sum_{k=0}^{2m}{4j\choose 2k}{2j-k\choose 2m-k}={2n+2m+1\choose 4m+1}2^{4m-1}$
1
vote
3answers
64 views

Calculate $\sum_{k=0}^n k \binom{n}{k} p^k (1-p)^{n-k}$

For $p \in [0,1]$ calculate $$S =\sum_{k=0}^n k \binom{n}{k} p^k (1-p)^{n-k}.$$ Since $$ (1-p)^{n-k} = \sum_{j=0}^{n-k} \binom{n-k}{j} (-p)^j, $$ then $$ S =\sum_{k=0}^n \sum_{j=0}^{n-k} k ...
10
votes
3answers
287 views

Prove that $\,\,\displaystyle\inf_{n\in\mathbb N}\sum_{k=0}^{p}\lvert\sin{(n+k)^p}\rvert>0$

For any poistive integer number $p$, show that $$\inf\left\{ {\left\vert\sin{(n^p)}\right\vert+\left\vert\sin{(n+1)^p}\right\vert+\cdots+ \left\vert\,\sin{(n+p)^p}\right\vert\, :\,n\in ...
4
votes
2answers
76 views

Series of inverses of binomial coefficients

Can you think of a simple way of proving that $$ \sum_{n=k+1}^\infty \frac{1}{n \choose k} $$ is rational for any $k \geq 2$? Here's the background. Consider a series: $$ \sum_{n=1}^\infty ...
1
vote
3answers
95 views

Understanding $n \left(\frac{2n \choose n}{4^n}\right)^2$ for large $n$

My answer over at cstheory.stackexchange.com involved the expression $$\lim_{n\to \infty} n \left(\frac{2n \choose n}{4^n}\right)^2$$ According to Wolfram Alpha, this expression is at most ...
2
votes
1answer
54 views

Maximum of a sequence $\left({n\choose k} \lambda^k\right)_k$

Is there an expression for the maximum of a sequence $\left({n\choose k} \lambda^k\right)_k$ (i.e. $\max_{k\in\{0,\ldots,n\}}{n\choose k}\lambda^k)$ in terms of elementary functions of $n$ and ...
8
votes
2answers
163 views

Ordinary generating function for $\binom{3n}{n}$

The ordinary generating function for the central binomial coefficients, that is, $$\displaystyle \sum_{n=0}^{\infty} \binom{2n}{n} x^{n} = \frac{1}{\sqrt{1-4x}}$$ follows from the generalized ...
1
vote
5answers
75 views

Calculate $\lim_{n\to\infty}\binom{2n}{n}2^{-n}$

I would like to show that: $$\lim_{n\to\infty}\binom{2n}{n}2^{-n} = \infty$$ I have gotten as far as: $$ \binom{2n}{n}={(2n)!\over (n!)^2}=({n\over1}+1)({n\over2}+1)(\dots)({n\over n}+1)\ge2^n $$ But ...
1
vote
2answers
140 views

Can I get a little help proving equality between a summation and integral?

Prove$$\sum_{k=0}^x \binom{n}{k}p^{k}(1-p)^{n-k} =(n-x)\binom{n}{x}\int_{0}^{1-p}t^{n-x-1}(1-t)^{x}dt.$$ Can someone show me the steps please? Here is the hint my book gave me: "Integrate by parts ...
4
votes
1answer
226 views

Proving $\binom{2n}{n}\ge\frac{2^{2n-1}}{\sqrt{n}}$

Prove that $$\binom{2n}{n}\ge\dfrac{2^{2n-1}}{\sqrt{n}}$$ By the way: I have see $$\binom{2n}{n}\ge\dfrac{4^n}{2n}=\dfrac{2^{2n-1}}{n}$$ proof: Applying the binomial theorem ...
0
votes
0answers
64 views

Binomial coefficient transformation

How can I write \begin{align*} \frac{1}{k} {N \choose k} \end{align*} as sum/difference of binomial coefficients without having $k$ in the denominator? Eventually, I can have multiple of $N$. In ...
8
votes
3answers
243 views

Sum of squares of binomial coefficients

I came across the following sum in reference to this question $$\sum_{n=0}^{\infty} \frac{1}{2^{5 n}} \binom{2 n}{n}^2 = \frac{\sqrt{\pi}}{\Gamma \left( \frac{3}{4}\right)^2}$$ The sum on the left ...
0
votes
1answer
374 views

What 's the upper limit of a binomial expansion with fractional power?

It's known that a binomial expansion can be writen as a sum, $\displaystyle (a+b)^n=\binom{n}{0}a^n+ \binom{n}{1} a^{n-1}b+\binom{n}{2}a^{n-2}b^2+.....$ If the power, $n$, is a natural number, the ...
0
votes
1answer
51 views

How to maximize $n!\sum^n_{k = 0}\frac{a^k(1+(-1)^{n-k})}{k!(n-k)!} \pmod{a^2}$ for a given $a$?

Short Version of the Question: How do I maximize the value of $n!\sum^n_{k = 0}\frac{a^k(1+(-1)^{n-k})}{k!(n-k)!} \pmod{a^2}$ for a given $a$? Long Version of the Question: I'm currently attempting ...
1
vote
0answers
57 views

Binomial Expansion problem error

I tried solving this question but failed. a) Expand $(1+2x)^{1/4}$ in ascending powers of $x$ up to and including the term in $x^3$, simplifying each term as far as possible. b) By substituting ...
8
votes
1answer
361 views

Summation of an Infinite Series: $\sum_{n=1}^\infty \frac{4^{2n}}{n^3 \binom{2n}{n}^2} = 8\pi G-14\zeta(3)$

I am having trouble proving that $$\sum_{n=1}^\infty \frac{4^{2n}}{n^3 \binom{2n}{n}^2} = 8\pi G-14\zeta(3)$$ I know that $$\frac{2x \ \arcsin(x)}{\sqrt{1-x^2}} = \sum_{n=1}^\infty ...
8
votes
1answer
185 views

Evaluate $\lim_{n \to \infty} \sum_{j=0}^{n}{{j+n-1} \choose j}\frac{1}{2^{j+n}}$

Evaluate $$\lim_{n \to \infty} \sum_{j=0}^{n}{{j+n-1} \choose j}\frac{1}{2^{j+n}}$$ I don't understand where to start. Please help.
5
votes
2answers
78 views

Prove that $\lim_{n \to \infty} \binom{n}{k}a^n = 0$

I'm working with this problem but I have no idea how to solve it. Here $k$ is fixed and $0<a<1$. I was trying to use that $\lim_{n \to \infty} a^n =0$ and that $\binom{n}{k}\leq\frac{n^k}{k!}$ ...
5
votes
2answers
145 views

Generalization Of The Binomial Theorem

Consider the sum $$\sum_{k=0}^{n_0} {n \choose k} \cdot \alpha^k$$ where $\alpha \in \mathbb{R}$ arbritary, $n_0 < n$. So it looks like binomial theorem, $$\sum_{k=0}^n {n \choose k} \cdot ...
3
votes
5answers
329 views

Spivak's Calculus - Exercise 4.a of 2nd chapter

4 . (a) Prove that $$\sum_{k=0}^l \binom{n}{k} \binom{m}{l-k} = \binom{n+m}{l}.$$ Hint: Apply the binomial theorem to $(1+x)^n(1+x)^m$. I'm having a hard time trying to solve the problem ...
1
vote
2answers
379 views

Limit of binomial coefficient

I would like to find the limit $$ \lim_{n \to \infty} \binom{s}{n+1} = \lim_{n \to \infty} \frac{s (s-1) \cdots (s-n)}{(n+1)!} , $$ where $s \in \mathbb C$. Actually, it would be enough to show that ...
1
vote
1answer
121 views

Help with binomial theorem related proof

I'm currently working through Spivak on my own. I'm stuck on this proof, and the answer key is extremely vague on this problem. I think I'm missing a manipulation involving sums. Prove that ...
1
vote
0answers
283 views

Calculation of a 'double' sum

Let $n \in N$ and $q\geq 2$. I am trying to calculate the following sum: $$ \sum_{i=0}^{\sqrt n/2}\sum_{j= i \sqrt n }^{(i+1)\sqrt n}\frac{(-1)^q2^q(\frac{n}{2}-j)^q}{(n-j)!j!} $$ Any help will be ...
7
votes
5answers
267 views

The $n^{th}$ root of the geometric mean of binomial coefficients.

$\{{C_k^n}\}_{k=0}^n$ are binomial coefficients. $G_n$ is their geometrical mean. Prove $$\lim\limits_{n\to\infty}{G_n}^{1/n}=\sqrt{e}$$
9
votes
4answers
214 views

Bounding ${(2d-1)n-1\choose n-1}$

Claim: ${3n-1\choose n-1}\le 6.25^n$. Why? Can the proof be extended to obtain a bound on ${(2d-1)n-1\choose n-1}$, with the bound being $f(d)^n$ for some function $f$? (These numbers ...
35
votes
4answers
2k views

$n$th derivative of $e^{1/x}$

I am trying to find the $n$'th derivative of $f(x)=e^{1/x}$. When looking at the first few derivatives I noticed a pattern and eventually found the following formula $$\frac{\mathrm d^n}{\mathrm ...