The Star Trek Problem in Williams's Book This problem is from the book Probability with martingales by Williams. It's numbered as Exercise 12.3 on page 236. It can be stated as follows:

The control system on the starship has gone wonky. All that one can do is to set a distance to be traveled. The spaceship will then move that distance in a randomly chosen direction and then stop. The object is to get into the Solar System, a ball of radius $r$. Initially, the starship is at a distance $R>r$ from the sun.
If the next hop-length is automatically set to be the current distance to the Sun.("next" and "current" being updated in the obvious way). Let $R_n$ be the distance from Sun to the starship after $n$ space-hops. Prove $$\sum_{n=1}^\infty \frac{1}{R^2_n}<\infty$$
holds almost everywhere.

It has puzzled me a long time. I tried to prove the series has a finite expectation, but in fact it's expectation is infinite. Does anyone has a solution?

in the book it seems to me  that the sum terminates when they get home, so if they get home the sum is finite.  williams also hints: it should be clear what thm to use here.  i am sure he is referring to levy's extension of the borel-cantelli lemma, $\frac 1 {R_n^2}$ is the conditional probability of getting home at time $n+1$ given your position at time n is $R_n$, it is the proportion of  the area of the sphere of radius etc.
