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How can I calculate the following sum?

any hint please? I am beginner in such things, i have tried to make a decomposition for the denominator but no result $$\sum_{n=0}^\infty \frac{1}{1+n^2}=?$$

amWhy
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F.Marko
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    Do you really need to calculate it, or just show that it converges? – bkarpuz Jul 22 '18 at 17:29
  • @bkarpuz yes i need to calculate it – F.Marko Jul 22 '18 at 17:30
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    The analytic approach is not for beginners. But you can get a close approximation with $$\sum_{n=0}^k \frac{1}{1+n^2} +\int_{k+1/2}^\infty \frac{1}{1+x^2}dx = \sum_{n=0}^k \frac{1}{1+n^2} +\frac{\pi}{2} - \tan^{-1}\left(k+\frac12\right)$$ for reasonable $k$ of say $100$ or more – Henry Jul 22 '18 at 17:45
  • There's also an answer here: https://math.stackexchange.com/questions/1110872/how-to-prove-sum-n-0-infty-frac11n2-frac-pi12-frac-pie – kobe Jul 22 '18 at 18:15

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