Rationality of subseries Let $(a_n)_n$ be a real sequence. Assume than $\sum_{n=1}^\infty a_n$ converges to a rational number.  Does there exist a subseries which converges to an irrational constant?
Assume now the opposite situation, I.e. the full series converges irrationally, can we find a (non finite) subseries converging to a rational number?
My guess is yes for both the questions, but I couldn't find any actual example. As for the first one, maybe I could extract a Liouville subseries from the geometric series of reason $10^{-1}$?
 A: As explained in this post, we could build a series such that every subseries has an irrational sum.
Conversely, if we have a series such that all but finitely many terms are zero and the rest are rational, then the sum of every subseries will be rational.
However, if we have a convergent series $\sum a_n$ such that infinitely many $a_n$ are non-zero, we may always select a subseries that converges to an irrational constant.  To prove this, we begin by constructing a subsequence: 


*

*Take $a_{n_1}$ to be non-zero.

*For each $k\geq 1$, take $a_{n_{k+1}}$ to be non-zero, satisfying
$$
|a_{n_{k+1}}| < 3^{-k} \cdot |a_{n_k}|
$$


Now, for any sequence $(\xi_k) \in \{0,1\}^{\Bbb N}$, we note that the map
$$
\Phi:(\xi_k) \mapsto \sum_{k=1}^\infty \xi_k \, a_{n_k}
$$
is injective (one-to-one), and that each $\sum_{k=1}^\infty \xi_k \, a_{n_k}$ is the sum of some (absolutely convergent) subseries.
Since the set $\{0,1\}^{\Bbb N}$ is uncountable and $\Phi$ is injective, we may conclude that there are at least uncountably many values that the sum of subseries may attain.  However, there are only countably many rationals.
So, there must exist a subseries whose sum is irrational.
