Tossing until heads and tails are level

A fair coin is tossed until, for the first time, the number of heads equals the number of tails (for example, $HHTHTT$ stops after $6$ tosses). Let $\mu$ be the expected number of tosses. What is $1/\mu$?

$0$. The difference of heads and tails is a simple symmetric random walk, and the stopping time is its first return to 0. That return is certain but has infinite mean (its tail decays like $n^{-1/2}$), so $\mu=\infty$ and $1/\mu=0$.