New posts in Summation

Prove $\frac{n}{\sum_{k=1}^n{\frac{1}{\frac{1}{k}+a_k}}}-\frac{n}{\sum_{k=1}^n{\frac{1}{a_k}}}\geqslant \frac{2}{n+1}$

Proving $\sum_{k=1}^n\frac{\sin(k\pi\frac{2m+1}{n+1})}k>\sum_{k=1}^n\frac{\sin(k\pi\frac{2m+3}{n+1})}k$

Hard power series problem

Prove $\sum_{k=0}^n\binom{n}{k}\cos\left(\frac{k\pi}{2} - \frac{n\pi}{4}\right) = \sqrt{2^n}$

Can this be proved using Combinatorics or generating functions?

Proving identity in infinite series

Trying to understand a double summation formula

Proving Riordan's identity that $\sum_{k=1}^{n} {n-1 \choose k-1} \frac{k!}{n^k}=1$

$Q\le \prod \frac{5+2x}{1+x}\le P$ find $P,Q$

Prove that $\sinh{2u}+2\sinh{4u}+3\sinh{6u}+...+n\sinh{2nu}=\frac{n\sinh{(2n+2)u-(n+1)\sinh{2nu}}}{4\sinh^2{u}}$