New posts in summation

Simplify binomial sum

summation binomial-coefficients eulerian-numbers

Looking for a closed form for $\sum_{k=1}^{\infty}\left( \zeta(2k)-\beta(2k)\right)$

sequences-and-series summation riemann-zeta pi dirichlet-series

How can we show that $ \sum_{n=0}^{\infty}\frac{2^nn[n(\pi^3+1)+\pi^2](n^2+n-1)}{(2n+1)(2n+3){2n \choose n}}=1+\pi+\pi^2+\pi^3+\pi^4 ?$

sequences-and-series summation power-series

Conjecture $\sum_{n=1}^\infty\frac{\ln(n+2)}{n\,(n+1)}\,\stackrel{\color{gray}?}=\,{\large\int}_0^1\frac{x\,(\ln x-1)}{\ln(1-x)}\,dx$

integration sequences-and-series definite-integrals summation closed-form

Evaluate integral: $\int_0^{\frac{\pi}{2}}\ln(a^2\cos^2 x +b^2\sin^2x)dx$?

definite-integrals summation

Extracting an asymptotic from a sequence defined by a recurrence relation

sequences-and-series summation asymptotics recurrence-relations experimental-mathematics

Proof of a binomial identity $\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$

combinatorics summation proof-writing binomial-coefficients taylor-expansion

Can someone explain me this summation?

discrete-mathematics summation

Generating function for cubes of Harmonic numbers

summation closed-form

Calculate the binomial $(1-x)^{-(n+1)}$

summation binomial-coefficients

A Binomial Coefficient Sum: $\sum_{m = 0}^{n} (-1)^{n-m} \binom{n}{m} \binom{m-1}{l}$

combinatorics summation binomial-coefficients

For any positive integer $n$, show that $\sum_{d|n}\sigma(d) = \sum_{d|n}(n/d)\tau(d)$

elementary-number-theory summation divisor-sum multiplicative-function divisor-counting-function

How find this $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}\zeta_{n}(3)}{n}=?$

summation

How can I prove, that this formula is related to the binomial series?

combinatorics summation binomial-coefficients

Why is this sum zero?

summation

How to prove that $\sum_{k=1}^n\frac{1}{\sqrt[n]{k!} }\sim \frac{n}{\ln n}$

sequences-and-series limits summation

Assymptotics of the generalized harmonic number $H_{n,r}$ for $r < 1$

summation asymptotics harmonic-numbers

Prove $\sum_{n=1}^\infty(e-\sum_{k=0}^n\frac1{k!})=1$

summation recreational-mathematics exponential-function factorial

For what $n$ can $\pm 1\pm 2\pm 3 ... \pm (n-1) \pm n = n+1$?

arithmetic summation

Different ways to come up with $1+2+3+\cdots +n=\frac{n(n+1)}{2}$

integration summation education telescopic-series