$\frac1n\sum _{k=1}^na_k\to0$ if and only if $\frac1n\sum _{k=1}^na^2_k\to0$
130693
Sequence such that $\lim\limits_{n\to \infty} \frac{x_1^2+x_2^2+...+x_n^2}{n}=0$
121723
$\phi(n)=\frac{n}{2}$ if and only if $n=2^k$ for some positive integer k
128415
Prove that $1<\frac{1}{1001}+\frac{1}{1002}+\frac{1}{1003}+\dots+\frac{1}{3001}<\frac43$
126105
Compute $\sum_{k=1}^{n} \frac 1 {k(k + 1)} $
37007
Evaluating $\sum_{k=1}^{n} \frac{1}{k(k+1)}$
61850
Compute $\sum_{k=0}^{\infty}\frac{1}{2^{k!}}$
25981
Prove that $\sum_{k=0}^n a_k x^k = 0$ has at least $1$ real root if $\sum_{k=0}^n \frac{a_k}{k+1} = 0$
29524
Formula for the sum $\sum_{i=2}^{n} \frac1{i^2-1}$
74117
Find the sum of series $\sum_{n=0}^{\infty}\frac{z^{kn}}{(kn)!}$
35381
Sum with binomial coefficients: $\sum_{k=1}^m  \frac{1}{k}{m \choose k} $
106028
Sum of $\frac{1}{(2n)(2n+1)}$ from $1$ to infinity
132182
$\sum_{n=1}^\infty \sum_{m=1}^\infty\frac{1}{(m+n)^2}$
89367
Calculate $ S =\sum_{k=1}^n\frac {1}{k(k+1)(k+2)}. $
25395
Sum $\sum_{n=1}^\infty \frac{n^2}{(n+2)!}$
66568
Number Theory : Show that $\sum_{i=0}^\infty$ $ [\frac{n}{2^i}+\frac{1}{2}]$ $=$ $2n$
16143
How can I prove that $\sum_{n=1}^\infty \frac{1}{n(n+1)} = 1$?
133955
Compute: $\sum_{k=1}^{\infty}\sum_{n=1}^{\infty} \frac{1}{k^2n+2nk+n^2k}$