Proofs of $\sum_{n=1}^\infty \frac{1}{n^n}=\int_0^1 \frac{1}{x^x}dx$
24508
Series as an integral (sophomore's dream)
71162
Showing that $\sum_{n=1}^{\infty} \frac{z^n}{1+z^{2n}}$ converges
91784
Calculating $\sum_{n=1}^\infty {\frac{1}{2^nn(3n-1)}}$
132182
$\sum_{n=1}^\infty \sum_{m=1}^\infty\frac{1}{(m+n)^2}$
17520
About the divergence of $\sum^\infty_{n=1}\frac 1 {n\cdot n^{1/n}}$
12423
"Closed" form for $\sum \frac{1}{n^n}$
16775
Showing that $\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$
16177
Proof by Induction: $\sum_0^nx^i=(1-x^{n+1})/(1-x)$
17332
How to compute $\sum\limits_{n=1}^{\infty}\frac{1}{x^n-1}$?
134152
Prove $\sum_{n=1}^\infty(e-\sum_{k=0}^n\frac1{k!})=1$
61547
Derive an explicit formula for a power series $\sum_{n=1}^\infty n^2x^n$
141892
How to evaluate $\int_0^\infty \frac{1}{x^n+1} dx$
13747
Proof of $\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}$
36366
Prove $\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}$
66568
Number Theory : Show that $\sum_{i=0}^\infty$ $ [\frac{n}{2^i}+\frac{1}{2}]$ $=$ $2n$
125341
Does $\sum_{n=1}^{\infty} \frac{1}{n+1}$ converge?
6416
How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$?