Using partial fractions: $$ \sum\limits_{k=0}^\infty \frac{1}{n^2+1} = \lim_{n \to \infty} \left(\frac{i}{2} \sum\limits_{k=0}^n \frac{1}{k+i} - \sum_{k=0}^n \frac{1}{k-i} \right) $$ Expanding and rationalising end up with: $$ \lim_{n \to \infty} \frac{i}{2}\left( -i + \frac{1-i}{1^2+1} + \cdots + \frac{n-i} {n^2+1} +\left( -i \frac{1+i}{1^+1} -\cdots- \frac{n+i}{n^2+1} \right)\right) $$ Simplifying I end up with an expanded version of what I started with: $$ \lim_{n \to \infty} \left( 1 +\frac{1}{1^2+1} + \frac{1}{2^2+1} + \cdots + \frac{1}{n^2+1}\right) $$ How can I solve this problem? I feel like I am just going in circles.
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$$\frac{1}{2} (1+\pi \coth (\pi ))$$ – David G. Stork Dec 25 '20 at 19:52
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https://math.stackexchange.com/questions/845506/series-expansion-of-coth-x-using-the-fourier-transform – Jean-Claude Arbaut Dec 25 '20 at 19:55
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https://math.stackexchange.com/questions/314986/value-of-sum-k-1-infty-frac1k2a2 – Etemon Dec 25 '20 at 19:57