Second red draw from a random urn as urns multiply

There are $n$ urns. Urn $i$ ($1\le i\le n$) holds one red ball and $2^i-1$ blue balls. You pick an urn uniformly at random and draw a ball, which turns out red. You put it back and draw again from that same urn. Let $p(n)$ be the probability the second ball is also red. What is $\lim_{n\to\infty}p(n)$?

$1/3$. By Bayes, $p(n)=\dfrac{\sum_i 4^{-i}}{\sum_i 2^{-i}}$ (the $1/n$ factors cancel). As $n\to\infty$ the sums tend to $\tfrac13$ and $1$, so the limit is $\tfrac13$.