# Convergence of series $\sum_{n=1}^{\infty} \sin\left( \frac{n\pi}{6}\right)$

Decide if the series $\sum_{n=1}^{\infty} \sin\left( \frac{n\pi}{6}\right)$ converges or not.

I've tried to use the ratio test but I had no success. I don't see how other convergence tests could work.

Thanks for your help!

• Can you think of any two subsequences of $\{\sin(n\frac {\pi}6)\}_n$ that converge to disparate values? – abiessu Oct 17 '13 at 15:10
• @abiessu yes, but I don't get why is this important since I am talking about the serie – Giiovanna Oct 17 '13 at 15:31

## 1 Answer

Hint: What is $\lim_{n\to\infty}\sin\left(\frac{n\pi}{6}\right)$?

• It does not exists, but I don't see how this can help – Giiovanna Oct 17 '13 at 15:30
• @user2768645: If it doesn't exist, then it must not equal $0$... – Clayton Oct 17 '13 at 15:31
• Ok, is this a criterion of convergence? – Giiovanna Oct 17 '13 at 15:35
• It is the first test for convergence taught in most calculus courses. – Clayton Oct 17 '13 at 15:38
• I didn't know it. Thanks for your attention! – Giiovanna Oct 17 '13 at 15:44