We have a positive series $\displaystyle\sum^\infty_{n=1}a_n$. is the following series converge or diverge ?$$\displaystyle\sum^\infty_{n=1}\frac{a_n}{1+a_n^2}$$
$$\sum^\infty_{n=1}\frac{a_n}{1+a_n^2}=\sum^\infty_{n=1}\frac{1}{{\frac 1 {a_n}}+a_n}$$
Now it's easy to see that if $a_n$ either converges or diverges, the series will converge.
If $\sum a_n<\infty$ then $$ \sum\frac{a_n}{1+a_n^2}\le \sum a_n<\infty. $$ Suppose now that $\sum a_n$ diverges. If $a_n$ is bounded, that is, $a_n\le M$ for some $M$, then $$ \sum\frac{a_n}{1+a_n^2}\ge\frac{1}{1+M^2}\sum a_n=\infty. $$ If $a_n$ is unbounded anything can happen. For instance, if $a_n=n^p$, $p\ge0$, then $\sum\frac{a_n}{1+a_n^2}$ converges if and only if $p>1$.
