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How can I find the solution to the series $\sum_{n=1} ^\infty n^2(\cos(nx) + i \sin(nx))$

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    If by "solution" you mean "limit", then you can't -- this series diverges for all $x$. To see this, note that the terms are $n^2\mathrm{e}^{\mathrm{i}nx}$, so the absolute value of the terms diverges, instead of going to $0$ as it must if the series is to converge.2011-04-05
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    @joriki: I will diverge for real $x$ but might well converge for complex $x$.2011-04-05
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    @Fabian: Well I hope you converge again :-) But seriously: You're right, although that would be rather an unusual way to write it down if $x$ were meant to be complex.2011-04-05
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    Fabian is right, it converges for complex $x$ if $\Im(x)>0$. In that case, you can get the limit by writing the terms as $n^2\mathrm{e}^{\mathrm{i}nx}$, writing that as the second derivative of $-\mathrm{e}^{\mathrm{i}nx}$ with respect to $x$, summing the geometric series and then taking the second derivative. (To make it a bit easier, you can add the $n=0$ term, since it's $0$ anyway.)2011-04-05

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