# How to sum this series for \pi/2\pi/2 directly?

The sum of the series

can be derived by accelerating the Gregory Series

using Euler’s Series Transformation. Mathematica is able to sum $(1)$, so I assume there must be some method to sum the series in $(1)$ directly; what might that method be?

Variations of the sum of reciprocals of the central binomial coefficients have been well-studied. For example, this paper by Sprugnoli (see Theorem 2.4) gives the ordinary generating function of $a_k = \frac{4^k}{(2k+1)}\binom{2k}{k}^{-1}$ to be
Subbing in $t = 1/2$ says that