The series is $$\sum_{n=1}^{\infty}\frac{\sin(nx)}{n^2}x^3.$$
We know that every $f_n(x)$ is continuous in $\Bbb R$. I wanted to apply some methods that need the series converges uniformly: Weierstrass M-test, if each $f_n(x)$ is continuous and the series converges uniformly then the series is also continuous, Cauchy's test for the uniform convergence to the sequence of partial sums but the problem is that I can't (or don't know) bound any of the $|f_n(x)|$ or $|S_r(x)−S_k(x)|$, where $S_n(x)=\sum_{j=1}^{n}\frac{\sin(jx)}{j^2}x^3$.
So, those are some basic methods I used for this exercise. Any help?
Consider an interval $[a,b]$, $|x^3<A$ on this interval and $|f_n(x)|\leq {A\over n^2}$ on this interval. This implies that $f_n$ converges uniformly on $[a,b]$ so is continuous.
https://en.wikipedia.org/wiki/Normal_convergence