Probability of drawing an element from a countably infinite sequence Consider a sequence containing $A$ and $B$ where, starting at $n=0$, there are $2^n A$'s followed by $2^{n+1} B \ $'s, so the sequence begins
$$A, B, B, A, A, B, B, B, B, A, A, A, A, B, B, B, B, B, B, B, B ...$$
If we continue this sequence indefinitely, what is the probability of drawing an A by randomly selecting and element of this sequence?
If we consider the sub-sequences, then whenever we hit a patch of A's, the probability of drawing an A from that sub-sequence approaches $1/2$, but then when we hit a string of B's, it will approach and reach $1/3$, before starting back toward $1/2$. To me, this shows that the sequence of probabilities of drawing an A from the sub-sequences does not converge in the ordinary calculus sense. However, does it converge in some weaker probability sense that gives this question a well-defined answer?
 A: I'd post this response as a comment, but I don't have enough reputation to (why do new accounts require 50 reputation to comment? Shouldn't it be the other way around? But I digress...). So please forgive any lack of mathematical rigor, as these are just intuitive thoughts.
In terms of convergence in probabilistic sense, it's clear that there is no convergence almost surely - there is no point in the sequence after which the probabilities drawing an A from the subsequence doesn't eventually reach $1/2$ or $1/3$ again. I'd want to say that there is no convergence in probability either, for similar reasons, as the probabilities in the partial sums alternate between $1/2$ and $1/3$.
At first glance, I'd want to say that the probability tends to $5/12$ when considering the distribution of partial sums in the sense of Ramanujan summation - in the long run, the number of subsequences where the probability of drawing an A in the partial sum is between $1/2$ and $5/12$ is the same as those between $5/12$ and $1/3$, and furthermore, the distribution of the probabilities is symmetric about $5/12$ (again, as n approaches infinity).
Not sure if this is correct, but it would be my hunch.
