I've been doing exercises with series for some time and now I've got an exercise that I'm supposed to solve using Abel or Dirichlet criteria. The problem is :
Determine if the following series is convergent or divergent by using Abel or Dirichlet criteria:
$$\sum_{n=1}^{\infty}\frac{ \sin{nx}}{n}$$
I do not know what to do, because:
First of all, Abel criteria says that if $\sum_{n=1}^{\infty}{x_n}$ is convergent and $y_n$ is a monotonous and bounded sequence, then $\sum_{n=1}^{\infty}{x_n y_n}$ is a convergent series, so in my case $\sin(nx)$ is bounded and monotonous, but $\sum_{n=1}^{\infty}{1 \over n}$ isn't convergent, so I think I can't use Abel cr.
Second, Dirichlet criteria says that the series $\sum_{n=1}^{\infty}{x_n}$ should have the sequence of partial sums bounded and $y_n$ should be decreasing and convergent to zero for $\sum_{n=1}^{\infty}{x_n y_n}$ to be convergent. In my case, whatever $\sum_{n=1}^{\infty}{x_n }$ I take it will not have bounded sequence of partial sums, so what should I do ? What should I use ? May be I am wrong somewhere ?
Thanks.