If $z_n \to a$, then $\frac 1n \sum_{k=1}^n z_k \to a$
13701
Prove convergence of the sequence $(z_1+z_2+\cdots + z_n)/n$ of Cesaro means
102942
Compute the sum $\sum_{k=1}^{\infty}k^mz^k$ where $|z|<1$
36083
If $k > 1$, then $\frac1{(k - 1)^2}-\frac1{(k + 1)^2}=\frac{4k}{(k^2 - 1)^2}$, hence simplify $\sum\limits_{k = 2}^n\frac k{(k^2 - 1)^2}$
74286
Prove: $\lim_{n\to\infty} \frac{\sum^n_{k=1} \frac{1}{k}}{ \ln(n) } = 1$
20317
Proving $\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$
74596
Coefficient growth in the power series $\sum u_n z^n = e^{1/(1-z)}$?
41792
If $abc=1$, then $\frac{a^{n+2}}{a^n+(n-1)b^n}+\frac{b^{n+2}}{b^n+(n-1)c^n}+\frac{c^{n+2}}{c^n+(n-1)a^n} \geq \frac{3}{n} $
9046
Prove that $\sum_{k=0}^{m}\frac{\binom{m}{k}}{\binom{n}{k}}=\frac{n+1}{n+1-m}$
14197
show that $\sum_{i=0}^l \dfrac{{l\choose i}}{{n\choose k + i}} = \dfrac{n+1}{(n-l+1) {n-l\choose k}}$
89728
Prove that $1<\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1}$
129801
Finding a closed form for $\sum^{n}_{k=1} \frac{k}{(k+1)!} $
150822
Finding $\sum_{k=0}^{n-1}\frac{\alpha_k}{2-\alpha_k}$, where $\alpha_k$ are the $n$-th roots of unity
349
Prove $| \sum_{i \leq n} \frac{\mu(i)}{i} | \leq 1$
139971
If $\sum{a_n}$ converges does that imply that $\sum{\frac{a_n}{n}}$ converges?
65993
Does $\sum_{k=1}^\infty\frac1{k^n}$ converge for $\Re(n)=1,\Im(n)\ne0$?
57503
How find $\sum_{k \in A} \frac{1}{k-1} $ for $ A = \{ m^n| \text{ } m, n \in Z \text { and } m, n \ge 2 \} $
30518
Formula for $\sum_{k=1}^n \frac{1}{k(k+1)(k+2)}$?