Newbetuts

If $\,a_n\searrow 0\,$ and $\,\sum_{n=1}^\infty a_n<\infty,\,$ does this imply that $\,n\log n\, a_n\to 0$?

Does $\Im(e^i+e^{e^i}+e^{e^i+e^{e^i}}\dots)$ converge?

How to compute $\lim\limits_{n\to\infty}\frac{1}{\sqrt{4n^2-1^2}}+\dots+\frac{1}{\sqrt{4n^2-n^2}}$.

Closed form for $\left(-1\right)^{n}\sum_{k=0}^{n}\left(-1\right)^{k}\binom{n}{k}2^{\binom{k}{2}}$

Show $\sum_{k=1}^{\infty}\left(\frac{1+\sin(k)}{2}\right)^k$ diverges

Mathematical induction for inequalities: $\frac1{n+1} + \frac1{n+2} + \cdots +\frac1{3n+1} > 1$

Asymptotics of the sum of squares of binomial coefficients

How to show that $ \left(1+\frac{1}{n} \right)^n = \sum_{i=0}^{n}\frac{1}{i!}\left(\prod_{j=0}^{i-1}\left(1 - \frac{j}{n}\right)\right)$ [duplicate]

Apéry's constant ($\zeta(3)$) value

What is the max of $n$ such that $\sum_{i=1}^n\frac{1}{a_i}=1$ where $2\le a_1\lt a_2\lt\cdots\lt a_n\le 99$?

Is there an exact solution for $\large\int \frac{dx}{\tan^{-1}(x)}$?

Prove the identity $\sum_{k=0}^{n}\sum_{r=0}^{k} \binom{k}{r} \binom{n}{k} = 3^n$

Prove: $\sum_{\sigma_1,\cdots,\sigma_N} e^{a\sigma_1}\cdots e^{a\sigma_N}=2\prod_{j=1}^{N}\sum_{\sigma_i}e^{a\sigma}$ where $\sigma_i \in \{-1,1\}$

Algebric proof for the identity $n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$

Combinatorial proof of $ \sum \limits_{i = 0} ^{m} 2^{n-i} {n \choose i}{m \choose i} = \sum\limits_{i=0}^m {n + m - i \choose m} {n \choose i} $

Can $n(n+1)2^{n-2} = \sum_{i=1}^{n} i^2 \binom{n}{i}$ be derived from the binomial theorem?

How to find $\sum_{r\ge 0} \binom{n}{r}\binom{n-r}{r} 2^{n-2r}$?

New posts in summation

Simplify $\sum_{i=0}^n (i+1)\binom ni$

Combinatorial argument for $\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$

Finding the sum of $1+4k+9k^2+...+n^2k^{n-1}$

If $\,a_n\searrow 0\,$ and $\,\sum_{n=1}^\infty a_n<\infty,\,$ does this imply that $\,n\log n\, a_n\to 0$?

Does $\Im(e^i+e^{e^i}+e^{e^i+e^{e^i}}\dots)$ converge?

How to compute $\lim\limits_{n\to\infty}\frac{1}{\sqrt{4n^2-1^2}}+\dots+\frac{1}{\sqrt{4n^2-n^2}}$.

Closed form for $\left(-1\right)^{n}\sum_{k=0}^{n}\left(-1\right)^{k}\binom{n}{k}2^{\binom{k}{2}}$

Show $\sum_{k=1}^{\infty}\left(\frac{1+\sin(k)}{2}\right)^k$ diverges

Mathematical induction for inequalities: $\frac1{n+1} + \frac1{n+2} + \cdots +\frac1{3n+1} > 1$

Asymptotics of the sum of squares of binomial coefficients

How to show that $ \left(1+\frac{1}{n} \right)^n = \sum_{i=0}^{n}\frac{1}{i!}\left(\prod_{j=0}^{i-1}\left(1 - \frac{j}{n}\right)\right)$ [duplicate]

Apéry's constant ($\zeta(3)$) value

What is the max of $n$ such that $\sum_{i=1}^n\frac{1}{a_i}=1$ where $2\le a_1\lt a_2\lt\cdots\lt a_n\le 99$?

Is there an exact solution for $\large\int \frac{dx}{\tan^{-1}(x)}$?

Prove the identity $\sum_{k=0}^{n}\sum_{r=0}^{k} \binom{k}{r} \binom{n}{k} = 3^n$

Prove: $\sum_{\sigma_1,\cdots,\sigma_N} e^{a\sigma_1}\cdots e^{a\sigma_N}=2\prod_{j=1}^{N}\sum_{\sigma_i}e^{a\sigma}$ where $\sigma_i \in \{-1,1\}$

Algebric proof for the identity $n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$

Combinatorial proof of $ \sum \limits_{i = 0} ^{m} 2^{n-i} {n \choose i}{m \choose i} = \sum\limits_{i=0}^m {n + m - i \choose m} {n \choose i} $

Can $n(n+1)2^{n-2} = \sum_{i=1}^{n} i^2 \binom{n}{i}$ be derived from the binomial theorem?

How to find $\sum_{r\ge 0} \binom{n}{r}\binom{n-r}{r} 2^{n-2r}$?