# Identify f(x)=\sum_{k=-\infty}^\infty\frac{1}{(x+k)^2}

Myah Fuller 2022-02-24 Answered
Identify
$f\left(x\right)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{{\left(x+k\right)}^{2}}$
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## Expert Answer

Athena Hussain
Answered 2022-02-25 Author has 5 answers
We can also use the well known summation formula

so we have to evaluate
$Re{s}_{z=-x}\frac{\pi \mathrm{cot}\left(\pi z\right)}{{\left(z+x\right)}^{2}}=\underset{z\to -x}{lim}\frac{d}{dz}\left(\pi \mathrm{cot}\left(\pi z\right)\right)=-{\pi }^{2}{\mathrm{csc}}^{2}\left(\pi x\right)$
hence
$\sum _{k\in \mathbb{Z}}\frac{1}{{\left(k+x\right)}^{2}}={\pi }^{2}{\mathrm{csc}}^{2}\left(\pi x\right)$
obviously if x is not an integer.
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