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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

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Don't use the title's body and the question's body as one, please. Type your question into the question body. –  Git Gud Mar 21 '13 at 18:55

2 Answers 2

up vote 8 down vote accepted

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.

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In the above limit would k be approaching infinity? I'm not quite sure how the limit equals 1 –  Fred Mar 21 '13 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 Tamaroff Mar 21 '13 at 19:22
    
Sorry, deleted my stupid comment...+1. –  1015 Mar 21 '13 at 19:23
    
@julien I'm curious! What did it say? =P –  Pedro Tamaroff Mar 21 '13 at 19:25
    
It said: what about $a_k=k\pi$? I had not read the very first line of your answer. –  1015 Mar 21 '13 at 19:26

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

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