I want to show that the following function converges uniformly and that it is continuous on $[T,-T]$
$f(x): = \Sigma_{n=1}^{\infty} \dfrac {\sin(x/n)}{n}$
Weierstrass crieteria:
We have $\vert \vert\dfrac {\sin(x/n)}{n}\vert\vert \leq \vert \vert \dfrac{\dfrac{T}{n}}{n} \vert \vert = T/n^2 \le \infty$ $\forall T \in \mathbb{R}$ Also $\Sigma T/n^2$ converges therefore $\Sigma_{n=1}^{\infty} \dfrac {\sin(x/n)}{n}$ converges uniformly
Therefore using Abel's criteria, $f(x)$ is conitnuous on $[T,-T]$
Is that the right way to do it?