Does a randomly chosen series diverge?

Question

Pick a point at random in the interval $[0,1]$, call it $P_1$.

Pick another point at random in the interval $[0,P_1]$, call it $P_2$.

Pick another point at random in the interval $[0,P2]$, call it $P_3$.

Etc...

Let $S = P_1+P_2+P_3+\cdots$

What is the probability that $S$ is divergent?

Any thoughts?

P.S. random, in this particular case, means equidistributed. I.e. $P(a<P_1<b)=b-a$.

Answer 1

For every $k$, $\mathbb E(P_k)=1/2^k$ thus: $$\mathbb E(S) = \mathbb E(P_1)+\mathbb E(P_2)+\cdots = 1/2 + 1/4 + \cdots= 1$$ Since $\mathbb E(S)$ is finite it follows that $P(S=\infty) = 0$, otherwise the expectation $\mathbb E(S)$ would be infinite.