Find the sum of the series using the Fourier series $\sum_{n=1}^\infty \frac{\sin(nx)}{n!}$. I think I should find a function that in the expansion in a Fourier series gives something similar on the formula above.
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Hint: $\sin (\sin (x)) (\sinh (\cos (x))+\cosh (\cos (x)))$. – David G. Stork Mar 08 '18 at 17:50
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1Perhaps this is also helpful: https://math.stackexchange.com/questions/1984531/how-do-i-find-the-sum-of-frac-cosnn/1984539#1984539 – imranfat Mar 08 '18 at 17:53
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HINT:
Note that $e^z=\sum_{n=0}^\infty \frac{z^n}{n!}$ and $\sin(nx)=\text{Im}\left((e^{ix)})^n\right)$. Now, let $z=e^{ix}$
Mark Viola
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and then $ \sum_{n=1}^\infty \frac{\sin(nx)}{n!} =$ Im $\sum_{n=1}^\infty \frac{e^{ix}}{n!}=$ Im $e^{e^z}$ and what is this? – user1223 Mar 08 '18 at 18:02
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$z=e^{ix}$. Hence, we have $$\text{Im}\left(e^{e^{ix}}\right)=\text{Im}\left(e^{\cos(x)+i\sin(x)}\right)$$Can you finish now? – Mark Viola Mar 08 '18 at 18:20
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