TITLE: Proving convergence for a series.
QUESTION [7 upvotes]: Let there be given that $\displaystyle\sum_{n=1}^{\infty}a_{n}$ is a convergent series and $a_{i}$ is not negative for every $i\in\mathbb{N}$.
And i want to prove that this series is also convergent (given above information):
$\displaystyle\sum_{n=1}^{\infty}\frac{n^2a_{n}+1}{(a_{n})^2+n^2}$.
So, i was thinking to transform it,by changing it's value into bigger and proving with "sandwich" criteria, but i am stuck at how to tranform it.
Any help would be very appreciated.

REPLY [14 votes]: Note that since $a_n\geq0$, you easily have $$\frac{n^2a_n+1}{n^2+a_n^2}= \frac{n^2a_n+1}{n^2(1+a_n^2/n^2)}\leq\frac{n^2a_n+1}{n^2}=a_n+\frac{1}{n^2}.$$ Now just use a comparison test.