Problem: Show that the following series $\sum\limits_{n=1}^{\infty}\frac{\sin(nx)}{{(n^4+x^4)}^{\frac{1}{3}}}$ is convergent.
It was expressly indicated that I should prove the series convergence in the following way:
Dirichlet´s test for Uniform convergence: The series converge uniformly
$\sum_\limits{n=k}^{\infty}f_ng_n$
on $S$ if $\{f_n\}$ converges uniformly to zero on $S$, $\sum(f_{n+1}-f_n)$ converges absolutely uniformly on $S$, and
$||g_k+g_{k+1}+...||_S\leqslant M\:\:n\geqslant K$
for some constant $M$.
Resolution attempt:
$f_n=\frac{1}{{(n^4+x^4)}^{\frac{1}{3}}}\to 0$ as $n\to\infty$
Now I must prove $\sum\limits_{n=1}^{\infty}{\sin(nx)}\leqslant M$
$\sin(nx)=\frac{e^{inx}-e^{-inx}}{2i}$
Question:
How can I use $\frac{e^{inx}-e^{-inx}}{2i}$ to major $\sum\limits_{n=1}^{\infty}{\sin(nx)}$?
Thanks in advance!