$\sum a_n$ converges $\implies\ \sum \sqrt{a_na_{n+1}}$ converges?
41836
If $\sum a_n$ converges, then $\sum \sqrt{a_na_{n+1}}$ converges
55914
If $\sum_{n=1}^{\infty} a_n$ is absolutely convergent, then $\sum_{n=1}^{\infty} (a_n)^2$ is convergent
127627
Does $\sum^{\infty}_{n=1} n/(n-1)!$ converge to something?
149806
Is it possible to utilize the convergence of the sequence $z_{n+1}=a/(1+z_n)$ to prove that the sequence $x_{n+2} = \sqrt{x_{n+1} x_n}$ is convergent?
139971
If $\sum{a_n}$ converges does that imply that $\sum{\frac{a_n}{n}}$ converges?
45818
Show that if $a_n$ is descending and $ \sum a_n$ is convergent then $na_n\to 0$
145863
sequence of positive numbers satisfying $a_{n+1}=\frac{2}{a_n+a_{n-1}}$, prove it converges
38522
Prove that if $\sum a_n$ converges absolutely, then $\sum a_{2n}$ converges.
73696
Given $(a_n)$ with $a_n \geq a_{n+1} \geq 0$ show that if $\sum \sqrt{a_n a_{n+1}} < \infty \implies \sum a_n < \infty$
116046
Prove that if $\sum_{n=1}^\infty a_{2n}$ and $\sum_{n=1}^\infty a_{2n-1}$ converge then $\sum_{n=1}^\infty a_n$ converges.
30471
Show that $a_{n+1}= 1 + \frac{1}{a_n}$ converges.
270
If $\sum_{n=1}^{\infty} a_n^{2}$ converges, then so does $\sum_{n=1}^{\infty} \frac {a_n}{n}$
139118
If $a_n$ goes to zero, can we find signs $s_n$ such that $\sum s_n a_n$ converges?
114141
If $\sum a_n^2 n^2$ converges then $\sum |a_n|$ converges
49419
Does $a_n$ converges if and only if $a_{2n},a_{3n},a_{2n-1}$ converge?
47405
sufficiency and necessity of convergence of $\sum a_n$ wrt convergence of $\prod (1 + a_n)$
158741
Convergence of $\sum \frac{a_n}{(a_1+\ldots+a_n)^2}$