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.

  • $\begingroup$ Don't ask questions by inserting a picture. Typeset them. $\endgroup$ – 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.

[$\sin (\frac 1 n +n \pi) =\sin (\frac 1 n) \cos(n\pi)+\cos (\frac 1 n) \sin (n\pi)=(-1)^{n} \sin (\frac 1 n)$]

  • $\begingroup$ Thank you!! I understand now! $\endgroup$ – 水野冬夜 Dec 3 '19 at 9:44

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