query: Proof that $\zeta {\space(3)} = \frac{5}{2} \sum{(-1)^{n-1}/{n^3}{2n\choose{n}}}$ and generalizations.
positive_passages.docid: 153231
positive_passages.text: Why does $\sum_1^\infty \frac1{n^3}=\frac52\sum_1^\infty\frac{(-1)^{n-1}}{n^3\binom{2n}{n}}$?
negative_passages.docid: 36023
negative_passages.text: Showing that $ \sum_{n=1}^{\infty} \arctan \left( \frac{2}{n^2} \right) =\frac{3\pi}{4}$
negative_passages.docid: 50435
negative_passages.text: Infinite Series $‎\sum\limits_{n=2}^{\infty}\frac{\zeta(n)}{k^n}$
negative_passages.docid: 6436
negative_passages.text: Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$
negative_passages.docid: 54432
negative_passages.text: What can be said about $\prod_{s=2}^{\infty} \zeta(s) $?
negative_passages.docid: 93813
negative_passages.text: How to show that $\sum\limits_{k=0}^n (-1)^k\tfrac{{ {n}\choose{k}}}{{ {x+k}\choose{k}}} = \frac{x}{x+n}$
negative_passages.docid: 112036
negative_passages.text: sum of series $\frac{n}{1\cdot 2 \cdot 3}+\frac{n-1}{2\cdot 3\cdot 4}+\frac{n-2}{3\cdot 4 \cdot 5}+\cdots \cdots n$ terms
negative_passages.docid: 96700
negative_passages.text: How is $\dfrac1{(1-x)^5}=\sum_{n\geq0}{n+4\choose4}x^n$
negative_passages.docid: 66380
negative_passages.text: Alternating harmonic sum $\sum_{k\geq 1}\frac{(-1)^k}{k^3}H_k$
negative_passages.docid: 16967
negative_passages.text: Sum : $\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^3}$
negative_passages.docid: 124135
negative_passages.text: A general formula for $\sum (k-1)(k-2)(k-3)$?
negative_passages.docid: 39524
negative_passages.text: Integral Representation of the Zeta Function: $\zeta(s)=\frac1{\Gamma(s)}\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$
negative_passages.docid: 67271
negative_passages.text: $\sum_{n=-\infty}^{\infty}\frac{1}{(u +n)^2}=\frac{\pi^2}{(\sin \pi u)^2}$
negative_passages.docid: 92440
negative_passages.text: Let $n \in \mathbb{Z}^+$, prove the identity $ \sum_{k=1}^{n-1} \binom {n} {k} \frac{kn^{n-k}}{k+1}=\frac{n(n^{n}-1)}{n+1}$
negative_passages.docid: 31867
negative_passages.text: Elementary proof that the limit of $\sum_{i=1}^{\infty} \frac{1}{\operatorname{lcm}(1,2,...,i)}$ is irrational
negative_passages.docid: 68421
negative_passages.text: Residue theorem with $\frac{1}{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) d\theta$.
negative_passages.docid: 113353
negative_passages.text: Double sum and zeta function