Check proof that if $a_{n+1} = \frac{1}{k}(a_n+\frac{k}{a_n})$ with $k>1$ and $a_1>0$, then $a_n\to\sqrt {\frac{k}{k-1}}$
110680
Convergence of recursive sequence $a_{n+1} =\frac{ 1}{k} \left(a_{n} + \frac{k}{a_{n}}\right)$
56680
$\sum_{n\geq 0}a_n$ converges iff $\sum_{n\geq 0} \frac{a_n}{\sum_{k=0}^n a_k}$ converges
3111
How to prove that if $\lim_{n \rightarrow \infty}a_n=A$, then $\lim_{n \rightarrow \infty}\frac{a_1+...+a_n}{n}=A$
73987
Prove the inequality $({1+\frac{a}b})^n$ + $(1+\frac{b}a)^n$ $\geq$ $2^{n+1}$
20317
Proving $\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$
112565
Prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{9}{a+b+c} : (a, b, c) > 0$
25981
Prove that $\sum_{k=0}^n a_k x^k = 0$ has at least $1$ real root if $\sum_{k=0}^n \frac{a_k}{k+1} = 0$
54659
How prove that $(a_{n})_{n \in \mathbb{N}}$ is convergent with $a_{0}=1$, $a_{n+1}= \sqrt{2a_{n}}$?
15205
Proving that the series 1 + ... + $1 / \sqrt{x}$ < $2 \sqrt{x}$
17289
prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit
12020
Prove $\sqrt{n+1}-\sqrt{n}\lt \frac{1}{2\sqrt{n}}$
108298
Prove $\sum_{i=1}^{i=n}\frac{a_i}{S-a_i}\geqslant \frac{n}{n-1}$ and $\sum_{i=1}^{i=n}\frac{S-a_i}{a_i}\geqslant n(n-1)$.
30471
Show that $a_{n+1}= 1 + \frac{1}{a_n}$ converges.
150088
Prove that $\sqrt {n − 1} +\sqrt {n + 1}$ is irrational for every integer $n ≥ 1$
17902
Prove or disprove: $\sum a_n$ convergent, where $a_n=2\sqrt{n}-\sqrt{n-1}-\sqrt{n+1}$.
106993
Use Complete Induction of set theory to prove $\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + ... + \frac{1}{\sqrt{n}} > \sqrt{n}$.
89728
Prove that $1<\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1}$