In this post we prove the formula for the sum of the reciprocals of the central binomial coefficients :

.

(Of course, the sum of the central binomial coefficients themselves does not converge.)

We start with the beta integral, .

Replacing *n* with and *k* with *n*, this becomes

.

Now, let and sum both sides from 0 to , moving the summation inside the integral. (We’re not performing a *u*-substitution here; the bounds and the differential remain in terms of *x*. The substitution is just to make the upcoming infinite series calculation easier to see.) We have

.

(The infinite series converges because *x* must be between 0 and 1, which means the maximum value of is 1/2.)

At this point we have the integral of a rational function. Partial fractions decomposition (or just rewriting the numerator as ) gets us to

.

Using trigonometric substitution, this evaluates to

.

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