To start out, I can demonstrate that any series of real numbers has a subseries that converges to an irrational.
Proof: Consider a sequence of positive real numbers, $a_n$, for whom $\sum a_n \in \mathbb R$. Because the sum of $a_n$ converges to a real then $a_n\to0$. Now we can take a subsequence of $a_n$ that converges as quickly as we like, so I'm going to find some subsequence of $a_n$, $b_n$, that has the property that $3b_{n+1} \le b_n$. Now every subsequence of $b_n$ has a 'unique' sum, in that no other subsequence of $b_n$ has the same sum as it. There are an uncountable number of subsequnces of $b_n$ and a countable number of rationals, thus some subsequence $b_n$, which in turn is a subsequence of $a_n$, must have a sum that converges to an irrational.
Now I've shown that every series of positive real numbers has a subseries that converges to an irrational number. My question is: "Is there an algorithm that would take in any series of real positive numbers that converges and would find a subseries that converges to an irrational number, and if there is such an algorithm what is it?"