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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1$\begingroup$ Don't use the title's body and the question's body as one, please. Type your question into the question body. $\endgroup$– Git GudMar 21, 2013 at 18:55
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2 Answers
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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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$\begingroup$ In the above limit would k be approaching infinity? I'm not quite sure how the limit equals 1 $\endgroup$– FredMar 21, 2013 at 19:12
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$\begingroup$ @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. $\endgroup$– Pedro ♦Mar 21, 2013 at 19:22
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$\begingroup$ It said: what about $a_k=k\pi$? I had not read the very first line of your answer. $\endgroup$– JulienMar 21, 2013 at 19:26
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use that $|\sin(y)|\leq |y| $ (prove with the series) else you could prove it with mean value theorem.
answered Mar 21, 2013 at 18:55