How can I tell if this serises converge or diverge?

Question

I need to check if this series converges absolutley:$$ \sum_{n=1}^{\infty}\frac{\sin(3n)}{\sqrt{4n+1}} $$

I showed it converge without the absolute value but didn't succeded for its absolute value:

$$\sum_{n=1}^{\infty}\frac{|\sin(3n)|}{\sqrt{4n+1}} $$

I don't know how to tell if it converges or diverges, I tried to use some comparing tests, but I didn't find it fruitful in that case.

Many thanks.

Answer 1

That series is not absolutely convergent. Note that$$|\sin(3n)|\geqslant\sin^2(3n)=\frac{1-\cos(6n)}2$$and that the series$\sum_{n=1}^\infty\frac{\cos(6n)}{2\sqrt{4n+1}}$ converges, by Dirichlet's test. However, the series $\sum_{n=1}^\infty\frac1{2\sqrt{4n+1}}$ diverges.