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One of my classmates challenged me to solve $\displaystyle\sum\limits_{n=1}^{\infty}\frac{\sin n}n=\;?$

With a simple c program I found that $\displaystyle\sum\limits_{n=1}^{1048576}\frac{\sin n}n\approx1.070796$. Later I found $\displaystyle1.070796\approx\frac{\pi-1}{2}$. My classmate told me I guessed right, but he ask me to prove it, and he gave me a hint that $\displaystyle e^{i\theta} = \cos\theta + i \sin\theta$, though I can't see the relationship between the question and the hint.

So how to prove $\displaystyle\sum\limits_{n=1}^{\infty}\frac{\sin n}n=\frac{\pi-1}{2}$?

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@Norbert While the accepted answer to that question answers this one, aren't they different questions? This one asks what the series converges to. All that that one asks is for a proof of convergnece. –  alex.jordan May 26 '13 at 6:36
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@Norbert The answer in your link confuses me. And what about my friend's hint that $\displaystyle e^{i\theta} = \cos\theta + i \sin\theta$? –  johnchen902 May 26 '13 at 6:44
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I wish my friends challenged me like that :( –  Prism May 26 '13 at 6:58
    
@johnchen902, if "your friend" really hinted you that $\,e^{i\theta}=\cos\theta+i\sin\theta\;$ and you did not know that before hand, it may be this exercise is not suited to your mathematics level. –  DonAntonio May 26 '13 at 8:42

1 Answer 1

up vote 24 down vote accepted

Make use of the following fact: For $\vert z \vert \leq 1$ and $z \neq 1$, we have $$\log(1-z) = - \sum_{k=1}^{\infty} \dfrac{z^k}k$$ Take $z=e^i$ and look at the imaginary part.

We hence have $$\log(1-e^i) = - \sum_{k=1}^{\infty} \dfrac{e^{ik}}k \implies \text{Imag}(\log(1-e^i)) = \text{Imag}\left(- \sum_{k=1}^{\infty} \dfrac{e^{ik}}k \right)$$ which gives us $$\sum_{k=1}^{\infty} \dfrac{\sin(k)}k = - \text{Imag}(\log(1-e^i))$$ $$\log(1-e^i) = \log \left(2\sin^2(1/2) - 2i \sin(1/2) \cos(1/2)\right) = \log(2\sin(1/2)e^{-i \pi/2}e^{i/2})$$ Hence, $$\text{Imag}(\log(1-e^i)) = \dfrac{1-\pi}2$$ which gives us $$\sum_{k=1}^{\infty} \dfrac{\sin(k)}k = \dfrac{\pi-1}2$$

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That, my friend, is an epic observation. :-) –  Parth Kohli May 26 '13 at 6:18

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