A second red ball from the same urn

There are $n$ urns; urn $i$ ($1 \le i \le n$) holds $1$ red ball and $2^i - 1$ blue balls (so $2^i$ balls total). You pick an urn uniformly at random, draw a ball, and it is red. You replace it and draw again from the same urn. Let $p(n)$ be the probability the second draw is also red. Compute $\displaystyle\lim_{n\to\infty} p(n)$.