# Values of x for which series can converge

I'm given a function $sin(nx)/(n^2)$ and I'm trying to find for which values of x the infinite series for this function would converge. It's easy to see that $sin(nx)$ is always between (-1,1), so then:

$-1/n^{-2} \le sin(nx)/n \le 1/n^{-2}$, so I could use the comparison test. But comparision test states that $a_n$ and $b_n$ must be positive, so would this series converge just for: $0 \le x \le$ $\pi/(2n)$ ?

Hint: consider series $|\sin nx|/n^2$; do they converge? Why does it imply convergence of original series?
• Yes that series converges by the comparison test as it is always less than $1/n^{2}$ but to use the comparison test, I have to indicate that $a_n$ is positive, right? So wouldn't that only be for [0, $\pi/2n$] ? – user180708 Nov 28 '14 at 10:50
• @user180708: this is indeed the case. If you want to check whether $\sum a_n$ converges, it is sufficient to check whether it converges absolutely, i.e. $\sum |a_n|<\infty$ – Ilya Nov 28 '14 at 10:54