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While I was was numerically integrating the magnetic field caused by an infinite array of magnetic moments, I observed the interesting limit ( limit (1) in the image). It may seem difficult to prove it mathematically but from the physics point of view, I think it can be proved relatively easily.

If we set i=0 ( no summation on i), the sum seems to have a limit too, but I have not not able find it in a close form ( for example in terms of pi). Any comments?

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