$\sum a_n$ converges iff $\sum \frac{a_n}{1+a_n}$ converges.
90696
Series in Real Analysis
100580
Proof that a sequence converges.
3744
If $\sum_{n=1}^{\infty} a_n$ converges, then so does $\sum_{n=1}^{\infty} a_n^{\frac{n}{n+1}}$
47405
sufficiency and necessity of convergence of $\sum a_n$ wrt convergence of $\prod (1 + a_n)$
74522
Convergence of a sequence, $a_n=\sum_1^nn/(n^2+k)$
96412
Prove $ \sum \frac{\cos n} { \sqrt n}$ converges
44261
Convergence of $\sum^\infty_{n=1} \ln(1+\frac 1 {2^n})$
14175
$\sum_\limits{n=0}^{\infty} a_n$ converges $\implies \sum_\limits{n=0}^{\infty} a_n^2$ converges
40797
$(a_n+a_{n+1})$ is convergent implies $(a_n/n)$ converges to $0$
126148
Convergence/divergence of $\sum\frac{a_n}{1+na_n}$ when $a_n\geq0$ and $\sum a_n$ diverges
142057
Prove that if $\sum_{n=1}^\infty a_n$ converges then $\sum_{k=1}^\infty A_k$ converges
128260
If $\sum a_n$ converges, so does $\sum a_n^{\frac{n}{n+1}}$
38522
Prove that if $\sum a_n$ converges absolutely, then $\sum a_{2n}$ converges.
91009
Convergence of the series $\sum a_n$ implies the convergence of $\sum \frac{\sqrt a_n}{n}$, if $a_n>0$
28166
If the series $\sum_0^\infty a_n$ converges, then so does $\sum_1^\infty \frac{\sqrt{a_n}}{n} $
135877
$\lim\limits_{n \to \infty} \frac{a_n}{b_n}=l$ ; Prove $\sum\limits_{k=1}^\infty a_k$ converges iff $\sum\limits_{k=1}^\infty b_k$ converges.
270
If $\sum_{n=1}^{\infty} a_n^{2}$ converges, then so does $\sum_{n=1}^{\infty} \frac {a_n}{n}$