Is $x_1=1$ and $x_{n+1}=\frac{1}{2}(x_n+\frac{2}{x_n})$ convergent?
128647
Calculating square roots using the recurrence $x_{n+1} = \frac12 \left(x_n + \frac2{x_n}\right)$
41751
$\sum_{n=1}^\infty \frac{i^n}{n}$ converge?
35885
Monotonicity and convergence of the sequence $a_n=\sum_{k=1}^{n}\frac{1}{k+n}$
110680
Convergence of recursive sequence $a_{n+1} =\frac{ 1}{k} \left(a_{n} + \frac{k}{a_{n}}\right)$
19159
Convergence of $\frac{1}{(\ln n)^{\ln n}}$
112430
Prove that sequence $ x_{(n+1)}= \frac {a}{1+x_n}$ is convergent to positive root of $x^2+x-a=0,$where $a >0$ and $x_1 >0$
68404
Check whether $\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }\frac{1}{\left(m+n\right)^2}$ converges or NOT?
9662
For what real values of $a$ does $\sum_{n=1}^{\infty}{(\frac{an}{n +1})^n}$ converge?
19225
Let $\{a_n\}$ be defined as follows: $a_1 = 2$, $a_{n+1}=\frac{1}{3-a_n}$, if $n \geq 1$. Does $\{a_n\}$ converge?
109718
Does $\sum \limits_{n=1}^\infty\frac{\sin n}{n}(1+\frac{1}{2}+\cdots+\frac{1}{n})$ converge (absolutely)?
30471
Show that $a_{n+1}= 1 + \frac{1}{a_n}$ converges.
141218
Does $\sum_{n=1}^{\infty} \frac{\sin(n)}{n}$ converge conditionally?
127251
$f_n(x) = \cos \left( \sqrt{x^2 + \frac{1}{n} }\right)$ is uniformly convergent
3744
If $\sum_{n=1}^{\infty} a_n$ converges, then so does $\sum_{n=1}^{\infty} a_n^{\frac{n}{n+1}}$
80432
Is the series convergent $x + x^{1+\frac{1}{2}}+x^{1+\frac{1}{2}+ \frac{1}{3}}+\cdots$
46830
Prove that the sequence $a_{n+1} =\frac{1}{2}\left(a_{n}+\frac{c}{a_{n}}\right)$ is convergent and find its limit
76867
Does $\sum _{k=2} ^\infty \frac{(-1)^k}{\sqrt{k}+(-1)^k}$ converge conditionally?