Let $f_n(x) = \frac{\sin(nx)}{n^2}$. I want to show that the infinite series $\sum f_n$ converges for all $x$.
After trying the ratio test and getting nowhere, I attempted to use the comparison test for absolute convergence using the series $\sum \frac{1}{n^2}$, which converges. Further,
$$0 \leq \frac{|\sin(nx)|}{n^2} \leq \frac{1}{n^2}$$
So $\sum f_n$ converges absolutely and therefore converges.
In determining the interval of convergence, I reasoned that because the above inequality is true for all real $x$, it must be that the interval of convergence is $\mathbb{R}$, but I'm not sure.
Would someone please show whether or not the above comparison test is sufficient for proving absolute convergence, and if my reasoning for the interval of convergence is also valid? Thank you.
Edit: my other concern is that the comparison test only works with series consisting of positive terms, yet my series has nonnegative terms. I don't think it's a problem, but I'm not sure about that either.