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## Homework Statement

Let ##r\in \mathbb{R}##. Determine whether ##\displaystyle \sum_{n=1}^{\infty}\sin \left(\frac{n\pi}{3}\right)\frac{1}{n^r}## converges

## Homework Equations

## The Attempt at a Solution

If ##r>1##, then ##|\sin \left(\frac{n\pi}{3}\right)\frac{1}{n^r}| \le |\frac{1}{n^r}|##. The latter converges, so by the direct comparison test, the original series converges.

Now I want to show that when ##-\infty < r \le 1## the series diverges, but I am having trouble figuring out how. Should I show that the limit of the terms is not zero?