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Tomorrow is Pi Day, 3/14 or 3-14 or even 3.14 using the "American" rather than "European" system of dates. The official time to celebrate is, of course, 15:19:26.535... using 24-hour (or what we Americans call military) time.

EDIT: Pi Day is cancelled. An extra digit sneaked in there somehow.

Sometimes people bake a special pie for Pi Day, with a circular crust whose circumference is Pi times its diameter, but I was wondering about the simple (though slow-to-converge) series

$$\frac\pi 4 = 1-\frac13+\frac15-\frac17+\cdots$$

What is the simplest, most direct and elementary proof that this series or sum does in fact converge to $\pi/4\,$?

EDIT: Possible duplicate: Yes, this is Leibnitz' result, based on the Taylor series for the arctangent, and it converges so slowly because it is on the very boundary or edge of the interval (or disk) of convergence -- but this is all mathematical commentary, because it is not too difficult to prove that it actually does converge, and provided it does, it does so to the correct value. I am especially curious about any possible proofs that do not involve calculus or trigonometry.


marked as duplicate by Parcly Taxel, Shaun, Winther, vrugtehagel, Professor Vector Mar 13 '18 at 19:54

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