Evaluating the function $S(a) = \sum_{n=0}^{\infty} a^n \binom{2n}{n}$.
The first thing that I noticed is that $S(a)$ converges when $a \in [-\frac{1}{4}, \frac{1}{4})$ because $$ \binom{2n}{n} \sim \frac{4^n}{\sqrt{\pi n}}. $$
However, this does not help me actually find $S(a)$. Furthermore, various strategies such as telescoping and DuSS (differentiation under the summation sign) have been foiled by the pesky binomial coefficient. If we switch to the Gamma function, DuSS is possible but extremely messy. I'm wondering what approach I should utilize in order to find an expression for $S(a)$ in terms of elementary functions.
A well-known formula is $$\sum_{n=0}^\infty\binom{2n}{n}x^n=\frac1{\sqrt{1-4x} }$$ for $|x|<1/4$. This can be verified by expanding the RHS via the binomial theorem.