# How to determine whether the series $\sum_{n=1}^\infty\sin (\frac{1}{n}+n\pi)$

Determine whether the series $$\sum_{n=1}^\infty\sin \left(\frac{1}{n}+n\pi\right)$$ is convergent or divergent.

How should I approach this question?

I've tried ratio test and comparison test, but it gives me weird answers.

I've tried comparing with y= 1/n for comparison test. But it gave me a value of 1?

Is this how you do it? Or is there another easier test that I can use?

Thanks for the help.

• Don't ask questions by inserting a picture. Typeset them. – Yves Daoust Dec 3 '19 at 9:25

The series is just $$\sum (-1)^{n} \sin (\frac 1 n)$$ and it is convergenet by the alternating series test.
[$$\sin (\frac 1 n +n \pi) =\sin (\frac 1 n) \cos(n\pi)+\cos (\frac 1 n) \sin (n\pi)=(-1)^{n} \sin (\frac 1 n)$$]