Transcendental subseries of a positive rational convergent series
= Transcendental subseries of a positive rational convergent series
Every convergent series of positive rational terms has a subseries whose rational partial sums $p_j/q_j$ satisfy
$$
0<\alpha-\frac{p_j}{q_j}<2^{-j}q_j^{-j}.
$$
Choose each new term small enough to enforce all previous tail bounds. Separation of reduced fractions first makes $\alpha$ irrational, and the <Liouville approximation theorem> then rules out every finite algebraic degree.