Im studying a book about stochastic processes and it is mentioned in the book that the equation below is true and we can obtain it by generating functions can anyone prove it. Thanks a lot. $$\sum_{m=0}^{\infty}\binom{2m}{m}p^m = (1 - 4p)^{-1/2}$$
How do you prove that $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$? I tried to identify the sum as a binomial series, but the $4$ and the $-1/2$ puzzle me. (This series arises in studying the first passage time of a simple random walk.)
Found the alternative form: $\sum _{k=1}^\infty ((1-\frac{1}{k})^{k})^k$. Tried various criteria, no luck so far.