Working on a random walk gambling problem, $a_n$ denotes a certain probability that a player wins a prize where the coin is fair. Variable $n$ denotes the number of the coin flip, and the probability of winning on an individual coin flip is $1/2$, and losing is also $1/2$.
Now, I have computed the correct individual probabilities as the following: $a_{2n} =\frac{1}{n+1}{2n\choose n}(1/2)^{2n}$.
The problem is, I need the sum of all these probabilities, because I need the probability of winning in the general case for all $n$, $n$ to $\infty$. I have no clue how to proceed. The Catalan numbers expand exponentially, but the $(1/2)^{2n}$ decays exponentially.
The generating function for Catalan numbers is $$C(x)=\sum _{n=0}^{\infty}C_nx^n=\frac{1-\sqrt{1-4x}}{2x},$$ this comes from the relation $c(x)=1+xc(x)^2.$
Notice that $x=1/4$ gives what you want.