Given the series
$$ \sum_{n=1}^\infty \frac{\sin(nx)}{n^3} $$ I have to show that the series converges both absolutely and uniformly on the real axis. I have shown without trouble that the series converges absolutely on the real axis but I am struggling to show that it converges uniformly on the real axis.
I know that a series converges uniformly if the partial sums $\{s_n\}$ where $s_n := \sum_{k=1}^n f_k$ converges uniformly but how do I prove that the partial sums converges uniformly? Do I have to use the definition of uniformly convergence saying that
$$ \forall \epsilon > 0 \exists N \in \mathbb{N} \forall x \in X \forall n \in \mathbb{N} : n \geq N \Rightarrow |f_n(x) - f(x)| \leq \epsilon $$ where $\{f_n\}$ is a sequence of functions on the set $X$ with the function $f: X \rightarrow \mathbb{C} $. Can you guide me in the right direction? Thanks in advance.
Use The M test of weistrass with $a_n=\frac{1}{n^3}$