How to prove the closed form $\left(\frac {1}{1-x}\right)^2 = \sum_{n=0}^{\infty}(n+1)x^n$
6416
How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$?
119606
Closed form of summation
50972
Prove that $\sum_{x=1}^{n} \frac{1}{x (x+1)(x+2)} = \frac{1}{4} - \frac{1}{2 (n+1) (n+2)}$
24407
How to prove the Fibonacci sum $\sum \limits_{n=0}^{\infty}\frac{F_n}{p^n} = \frac{p}{p^2-p-1}$
116019
Find the limit $\lim\limits_{n\to \infty}\sum\limits_{k=1}^{n}\left(\frac{k}{n^2}\right)^{\frac{k}{n^2}+1}$
145019
Prove that $\sum_{n=0}^\infty \frac{(-1)^n}{3n+1} = \frac{\pi}{3\sqrt{3}}+\frac{\log 2}{3}$
76756
A convergence proof: $\lim_{n\to\infty} \left(1+n^2\right)^{\frac1n}$
60098
How to compute $\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$?
114673
Proving the convergence of $\sum_{n=2}^{\infty} \frac{1}{n (\ln n)^2}$
4414
Computig the series $\sum\limits_{n=2}^\infty \ln\left(1-\frac{1}{n^2}\right)$
5272
Proving that $\sum_{k=0}^{n}\frac{(-1)^k}{{n\choose k}}=[1+(-1)^n] \frac{n+1}{n+2}.$
140614
Show that $\sum^\infty_{n=m}\frac{1}{n^2} = O(\frac{1}{m})$
30407
Prove that $\sum_{n=1}^\infty\frac{n}{n^2+1}$ is divergent
104853
Proving that $\lim\limits_{n\to\infty}n\left( \int_0^1 f(t)\, dt -\frac1n\sum_{k=0}^{n-1}f\left(\frac k n\right) \right)=\frac{f(1)-f(0)}{2}$
67752
Show that $\sum_{n = 1}^{+\infty} \frac{n}{2^n} = 2$
133160
Closed form for $\int_0^\infty\frac{1}{(1+x^2)^s}\,dx$ when $s\in (0.5,\infty)\setminus\mathbb{N}$
19914
Closed form for $\sum_{n=0}^{\infty} \binom{n+k}{k} x^n$ (Negative Binomial Theorem)