How to prove $\sum_{k=1}^∞ k r^{k-1} = \frac{(1)}{(1-r)^2}$
6416
How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$?
17157
Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction
106552
Find the sum of $\sum \frac{1}{k^2 - a^2}$ when $0<a<1$
24108
Proof of the equality $\sum\limits_{k=1}^{\infty} \frac{k^2}{2^k} = 6$
113585
Does $\sum\limits_{k=1}^∞\frac1{k^{\ln k}}$ converge or diverge?
34349
Prove that $\frac{1}{1+x_1}+\frac{1}{1+x_2}+\cdots+\frac{1}{1+x_n} \geq \frac{n}{\sqrt[n]{x_1x_2\cdots x_n}+1}$
104447
Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$
126105
Compute $\sum_{k=1}^{n} \frac 1 {k(k + 1)} $
30407
Prove that $\sum_{n=1}^\infty\frac{n}{n^2+1}$ is divergent
17714
Proof $\sum_{k=1}^n (-1)^{k+1} \binom{n}{k}\frac{1}{k} = H_n$ by induction
140614
Show that $\sum^\infty_{n=m}\frac{1}{n^2} = O(\frac{1}{m})$
154544
Prove that $ \sum_{n=0}^\infty \frac{z^{2^n}}{1-z^{2^{n+1}}}=\frac{z}{1-z} $
142985
Finding sum of series $\sum^{\infty}_{k=0}\frac{(k+1)(k+2)}{2^k}$
14197
show that $\sum_{i=0}^l \dfrac{{l\choose i}}{{n\choose k + i}} = \dfrac{n+1}{(n-l+1) {n-l\choose k}}$
106281
Prove $\lim\limits_{n\to \infty}\frac{1}{\sqrt n}\left|\sum\limits_{k=1}^n (-1)^k\sqrt k\right|= \frac{1}{2}$
58969
A proof for the inequality $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2} $ for all $t \geq 2$
36083
If $k > 1$, then $\frac1{(k - 1)^2}-\frac1{(k + 1)^2}=\frac{4k}{(k^2 - 1)^2}$, hence simplify $\sum\limits_{k = 2}^n\frac k{(k^2 - 1)^2}$