Let denote the sum of the first n terms of the series. Then is the same as the sum of the first 2n terms of the Leibniz series (because there is no problem with rearranging the terms in a finite sum). So converges to the same limit L as the Leibniz series.

Next, the (2n+1)th term in the series is , which converges to 0 as Therefore

Finally, since and converge to the same limit L, it follows that the whole series converges to L.