Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Are these series convergent or divergent?

$$ \sum_{}^\infty [\sin(\frac{n\pi}{6})]^n $$


$$\sum_{}^\infty [\sin(\frac{n\pi}{7})]^n $$

share|cite|improve this question
up vote 4 down vote accepted

The values of the sine cycle through a finite number of values. You can easily see that $$\limsup_{n\rightarrow\infty} \left|\sin\left(\frac{n\pi}{6}\right)\right| = 1$$ while $$\limsup_{n\rightarrow\infty} \left|\sin\left(\frac{n\pi}{7}\right)\right| < 1$$ This means the first sum cannot converge while the second sum will be absolutely convergent.

share|cite|improve this answer

Taking $\,n=3k\,\,,\,k\in\Bbb N\,\,,\,\,k\,\,\text{odd}$ , we get

$$\sin\frac{n\pi}{6}=\sin\frac{k\pi}{2}=\pm 1\Longrightarrow \sin^n\frac{n\pi}{2}\rlap{\;\;\;\;\;/}\xrightarrow [n\to\infty]{}0$$

so the series cannot converge.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.