Absolute and conditional convergence of a series with $\sin(x)$

Question

I have to explore absolute and conditional convergence of this function series

I tried to find $a(n)$ and $a(n+1)$ terms of the series and then divide it and take a limit. But I've got nothing. Uppermost I also have to find a module of a(n)/a(n+1). to explore absolute and conditional convergence. Please, help me solve this example.

Answer 1

Hint:

The $\;p$-series test tells us that

$$\sum_{n=1}^\infty\frac1{n^p}\;\;\;\text{converges iff}\;\;p>1\;\ldots$$

Besides the above: what conditional or whatever series? This is always a positive series.