Suppose $x_1 ≥ x_2 ≥ · · · ≥ 0$. Then $\sum_{n=1}^{\infty} x_{n}$ converges if and only if $\sum_{n=1}^{\infty} 2^{n}x_{2^n}$ converges.
140610
Cauchy condensation test
9921
Convergence of $\sum_{n=1}^\infty\frac{n}{(n+1)!}$
23894
$a_n>0$ and $(a_n)$ is decreasing. Suppose that $\sum_{n=1}^{\infty}{a_{2n}}$ converges. Prove that $\sum_{n=1}^{\infty}{a_n}$ also converges.
102012
Determine if $\sum_{n=2}^{\infty} \frac{(-1)^n}{n+(-1)^n}$ converges or diverges
25503
Convergence of $\sum_{n=1}^\infty \frac{\sin^2(n)}{n}$
35929
Convergence of series $\sum_{n=1}^{\infty}{\frac{1}{\sqrt{1}+\sqrt{2}+...+\sqrt{n}}}$
47740
Let $x_1=a>0$ and $x_{n+1}=x_n+\frac{1}{x_n} \forall n\in \mathbb N$. Check whether the following sequence converges or diverges.
49419
Does $a_n$ converges if and only if $a_{2n},a_{3n},a_{2n-1}$ converge?
20142
$\sum a_n$ be convergent but not absolutely convergent, $\sum_{n=1}^{\infty} a_n=0$
127627
Does $\sum^{\infty}_{n=1} n/(n-1)!$ converge to something?
28587
Show that $\sum\limits_{n=1}^\infty \frac {\sqrt{a_n}}{n}$ converges if $\sum\limits_{n=1}^\infty{a_n}$ does provided that $a_n>0$
10460
Why does $\sum_{n = 0}^\infty \frac{n}{2^n}$ converge to 2?
56680
$\sum_{n\geq 0}a_n$ converges iff $\sum_{n\geq 0} \frac{a_n}{\sum_{k=0}^n a_k}$ converges
98448
Find the radius of convergence of $\sum_{n=1}^{\infty}{n!x^{n!}}$
67303
If $\sum_{n=2}^{\infty} a_n$ is convergent, is $\sum_{n=2}^{\infty}{\sqrt{a_n} \over \ln{n}}\left( n^{a_n}-1 \right)$ convergent as well?
44288
Is my proof of convergence for $\sum\limits^{\infty}_{n=1} \frac {2^n} {n^n}$ true?
41751
$\sum_{n=1}^\infty \frac{i^n}{n}$ converge?