An Alternating Sum

Calculus Level 5

\[S=\sum_{n=-\infty}^{\infty} \dfrac{(-1)^n}{1+n^2}=\dfrac{a\pi e^{b\pi}}{e^{c\pi}-1},\]

where \(a,b,c \) are natural numbers. Find \(a+b+c\).

Note: Please avoid using wolfram alpha.

Hint: Think about the Fourier transform of \(f(x)=e^{-|ax|}\).

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