convergence of series of sinus functions

Question

Consider the series of functions

$$ \sum_{n=1}^{\infty} \frac{1}{n} \left( \frac{1}{2}\sin(x) \right)^n $$

Does this series converge simply/absolutely/uniformly/normally on $\mathbb{R}$?

My attempt:

1. I tried to prove that this series converges normally (which would imply the other 3), but I found that it wasn't because $ \sum_{n=1}^{\infty} \frac{1}{n} $ is divergent.

2. Then I found that it converges pointwise in $x=0+k\pi$ to the null function.

3. I also found that it isn't uniformly convergent.

I'm thinking this could be completely wrong, which is why I'm trying to get some help. Thanks in advance!

Answer 1

That series is normally convergent on $\Bbb R$: $\sup_{x\in\Bbb R}\left|\frac1n\left(\frac12\sin(x)\right)^n\right|=\frac1{n2^n}$ and the series $\sum_{n=1}^\infty\frac1{n2^n}$ converges.