# If series $\sum a_n$ is convergent with positive terms does $\sum \sin a_n$ also converge?

If $\{a_n\}$ is a sequence of positive terms such that the series $$\sum_{n=1}^\infty a_n$$ coverges, does the series $$\sum_{n=1}^\infty \sin a_n$$ also converge?

I believe that limit comparison test is necessary but I'm not sure how to use it here

• Don't use the title's body and the question's body as one, please. Type your question into the question body. Mar 21, 2013 at 18:55

Since $\lim\limits_{k \to \infty} a_k=0$, $$\tag 1\lim_{k \to \infty} \frac{\sin a_k}{a_k}=1$$

Thus $\sum_k |\sin a_k|$ converges $\iff \sum_k |a_k|=\sum_k a_k$ does. By your hypothesis and the above, $\sum_k \sin a_k$ will be absolutely convergent, so it will converge.

• In the above limit would k be approaching infinity? I'm not quite sure how the limit equals 1
– Fred
Mar 21, 2013 at 19:12
• @Michael Recall that $$\lim_{x\to 0}\frac{\sin x}x=1$$ Yes, as $k\to\infty$; $a_k\to 0$. Since $\sin$ is continuous, $(1)$ above follows.
– Pedro
Mar 21, 2013 at 19:22
• Sorry, deleted my stupid comment...+1. Mar 21, 2013 at 19:23
• @julien I'm curious! What did it say? =P
– Pedro
Mar 21, 2013 at 19:25
• It said: what about $a_k=k\pi$? I had not read the very first line of your answer. Mar 21, 2013 at 19:26

use that $|\sin(y)|\leq |y|$ (prove with the series) else you could prove it with mean value theorem.