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Help needed!!

I was just reading Feynman lectures and there was a series which came there. There its value was assumed. But I thought it must be having a value as it is converging. But it was really difficult to find a closed form. The series was -

$\displaystyle \sum_{k=0}^{\infty}{\frac{{(-1)}^{k}}{\sqrt{{x}^{2}+{k}^{2}}}}$

I learnt Fourier Transform and Poisson sum formula too but they are not helping me. Actually the topic was electrostatic energy of an ionic crystal.

Any type of help will be appreciated!

Note by Kartik Sharma
2 years ago

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Did you look up Ewald's summation ? As far as I know , fourier transform comes up in that summation only , and if that is indeed the case , sorry because I $$\in [Novice^{+},Intermediate^{-}]$$ ! · 2 years ago

:D Interesting interval there. · 2 years ago

Haha yes! · 2 years ago

Well, it came in one of Pratik's questions as well and that question is quite a bit same as this one. · 2 years ago