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Evaluate $$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$
where $i=\sqrt{-1}$

For this question, I did the following,

Let $$ \begin{align*} S &= \sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n} \\ C &= \sum_{n=1}^{\infty} \dfrac{\cos n}{i^n \cdot n} \end{align*} $$ We have to evaluate $|S|$ $$\implies S = \Im{(C+iS)}=\Im\displaystyle \sum_{n=1}^{\infty} \dfrac{e^{in}}{i^n \cdot n}$ $ However, due to the $n$ in the denominator, I cannot sum the series. If only it had been in the numerator I would've sum it as an A.G.P.

Can anyone suggest something?

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    $\begingroup$ aDo you know series for $\ln(1+z)$? $\endgroup$ – Robert Israel Jan 25 '15 at 19:00
  • $\begingroup$ See here. $\endgroup$ – Mhenni Benghorbal Jan 25 '15 at 20:36
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After a lot of struggle I found this:

$$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|=$$

$$S=\frac{1}{2}\left|-ln(1+ie^{-i})+ln(1+ie^i)\right|$$

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  • $\begingroup$ Can you please provide the details too? Thanks! $\endgroup$ – user208998 Apr 26 '15 at 17:18
  • $\begingroup$ I was struggling with this in school and solved it together with my prof. You've write the sum easier $\endgroup$ – Jan Apr 26 '15 at 17:20
  • $\begingroup$ But can you at least provide some hints? $\endgroup$ – user208998 Apr 26 '15 at 17:23
  • $\begingroup$ Use De Moivre and euler $\endgroup$ – Jan Apr 26 '15 at 17:25

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