Evaluate $$\sum_{n=0}^\infty\frac{1}{n^2+1}$$ how would I go to evaluate this? I've tried plugging a few terms, $$1+\frac{1}{2}+\frac{1}{5}+\frac{1}{10}+\frac{1}{17}+\frac{1}{26}+\dots$$ then writing a few sums: $$S_1=1, S_2=\frac32, S_3=\frac{17}{10}, S_4=\frac{18}{10},S_5=\frac{158}{85}$$ but I can't seem to be able to identify a pattern. There is no way I can wrap my head to approach series like this simple, any hints? Also Riemann's sums don't seem to be applicable here it seems.
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Does this answer your question? Find the infinite sum of the series $\sum_{n=1}^\infty \frac{1}{n^2 +1}$ – found with Approach0 – Martin R Oct 03 '21 at 17:33
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is there a less-advanced method maybe? – Acyex Oct 03 '21 at 17:41
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I doubt it. But this has been asked more than once, you can check https://math.stackexchange.com/questions/linked/736860 if there is an answer that suits you more. – Martin R Oct 03 '21 at 17:51
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This specific answer by J.d'Aurizio https://math.stackexchange.com/a/888690/399263 gives a method not using calculus in $\mathbb C$, but is it less-advanced ? At least it uses Fourier series in the real domain, as it is the case for the classic calculation of $\sum \frac 1{n^2}$. – zwim Oct 03 '21 at 18:03