Expected head fraction when stopping once ahead

You toss a fair coin as many times as you like and score the fraction of tosses that came up heads, i.e. the ratio $a:1$ of heads to total tosses. Suppose you stop the first time the number of heads exceeds the number of tails. What is the expected value of $a$, to three decimal places?

$\pi/4\approx0.785$. You stop at toss $2n+1$ with $n+1$ heads, which happens with probability $C_n/2^{2n+1}$ ($C_n$ Catalan). The expected ratio is $\sum_n \tfrac{C_n}{2^{2n+1}}\cdot\tfrac{n+1}{2n+1}=\tfrac12\sum_n\binom{2n}{n}\tfrac{4^{-n}}{2n+1}$, which is $\tfrac12\arcsin(1)=\tfrac{\pi}{4}$.