TITLE: How to determine whether or not this series converges?
QUESTION [6 upvotes]: $$\displaystyle\sum\limits_{n=1}^\infty \dfrac{n^{-1/2}}{2+\sin^2(n)}$$
I tried the divergence test but it approaches 0.
It's not a geometric series.
Doubt I could apply the integral test.
Not a p-series nor an alternating series.
Can't use root test.
I suppose I am left with either ratio test or a comparison test but I couldn't figure those out.
Advice?

REPLY [7 votes]: Hint:  Try using the comparison test and comparing to the function $\frac{1}{\sqrt{n}}$.  To apply this, notice that $2\leq 2+\sin^2(n)\leq 3$  so that $$\frac{n^\frac{-1}{2}}{3}\leq \frac{n^\frac{-1}{2}}{2+\sin^2(n)}\leq \frac{n^\frac{-1}{2}}{2}.$$