Newbetuts

Prove that $\sum_{q=0}^{d-r}\sum_{s=r+q}^{d}{{\binom {r-1+q}{r-1}}(r-1)!s}=\sum_{s=r}^{d}{\binom{s}{r}(r-1)!s}$ by sum manipulation [closed]

How find this sum $f(x)=\sum_{n=1}^{\infty}\frac{x^{n^2}}{n}$

Evaluating $\sum_{cyc} \frac{a^4}{(a-b)(a-c)}$, where $a=-\sqrt3+\sqrt5+\sqrt7$ , $b=\sqrt3-\sqrt5+\sqrt7$, $c=\sqrt3+\sqrt5-\sqrt7$

How prove this sum $\sum_{k=1}^{+\infty}(2k-1)!!\left(\binom{n-1}{2k-2}-n\binom{n-2}{2k-1}-\binom{n-2}{2k}\right)=1$

Prove $\sum_{k=0}^n \binom{n}{k}(-1)^k \frac{x}{x+k} = \prod_{k=1}^n \frac{k}{x+k}$ and more

Calculate the sum: $\sum_{x=2}^\infty (x^2 \operatorname{arcoth}(x) \operatorname{arccot} (x) -1)$

Double sum depending only on sum of arguments

Rolling a die - Conditional Probability

How to prove this inequality involving the minimum of two chosen numbers

Showing $\sum\limits^N_{n=1}\left(\prod\limits_{i=1}^n b_i \right)^\frac1{n}\le\sum\limits^N_{n=1}\left(\prod\limits_{i=1}^n a_i \right)^\frac1{n}$?

Summation over Weierstrass $\wp$ functions

Staver's identity relating $\sum_{k=1}^{n}\binom{2k}{k}\frac{1}{k}$ and $\sum_{k=1}^{n}\left(k\binom{n}{k}\right)^{-2}$

Notation: What is the scope of a sum?

$\sum\limits_{i=1}^n \frac{x_i}{\sqrt[n]{x_i^n+(n^n-1)\prod \limits_{j=1}^nx_j}} \ge 1$, for all $x_i>0.$

Closed form for Euler sum with $H_{2n}$?.

Help with binomial identity

What is the probability on rolling $2n$ dice that the sum of the first $n$ equals the sum of the last $n$?

New posts in summation

Double summation $\sum_{m,n=1\, m\neq n}^\infty{\frac{m^2+n^2}{mn(m^2-n^2)^2}}$

Finding sum : ${\mathop{\sum\sum\sum\sum}_{0\le i\lt j\lt k\lt l\le n }} \,1$

A combinatorial proof of the identity: $\sum_{k=1}^n k \binom{n}{k}^2 = {n}\binom{2n-1}{n-1}$?

Prove that $\sum_{q=0}^{d-r}\sum_{s=r+q}^{d}{{\binom {r-1+q}{r-1}}(r-1)!s}=\sum_{s=r}^{d}{\binom{s}{r}(r-1)!s}$ by sum manipulation [closed]

How find this sum $f(x)=\sum_{n=1}^{\infty}\frac{x^{n^2}}{n}$

Evaluating $\sum_{cyc} \frac{a^4}{(a-b)(a-c)}$, where $a=-\sqrt3+\sqrt5+\sqrt7$ , $b=\sqrt3-\sqrt5+\sqrt7$, $c=\sqrt3+\sqrt5-\sqrt7$

How prove this sum $\sum_{k=1}^{+\infty}(2k-1)!!\left(\binom{n-1}{2k-2}-n\binom{n-2}{2k-1}-\binom{n-2}{2k}\right)=1$

Prove $\sum_{k=0}^n \binom{n}{k}(-1)^k \frac{x}{x+k} = \prod_{k=1}^n \frac{k}{x+k}$ and more

Calculate the sum: $\sum_{x=2}^\infty (x^2 \operatorname{arcoth}(x) \operatorname{arccot} (x) -1)$

Double sum depending only on sum of arguments

Rolling a die - Conditional Probability

How to prove this inequality involving the minimum of two chosen numbers

Showing $\sum\limits^N_{n=1}\left(\prod\limits_{i=1}^n b_i \right)^\frac1{n}\le\sum\limits^N_{n=1}\left(\prod\limits_{i=1}^n a_i \right)^\frac1{n}$?

Summation over Weierstrass $\wp$ functions

Staver's identity relating $\sum_{k=1}^{n}\binom{2k}{k}\frac{1}{k}$ and $\sum_{k=1}^{n}\left(k\binom{n}{k}\right)^{-2}$

Notation: What is the scope of a sum?

$\sum\limits_{i=1}^n \frac{x_i}{\sqrt[n]{x_i^n+(n^n-1)\prod \limits_{j=1}^nx_j}} \ge 1$, for all $x_i>0.$

Closed form for Euler sum with $H_{2n}$?.

Help with binomial identity

What is the probability on rolling $2n$ dice that the sum of the first $n$ equals the sum of the last $n$?