TITLE: Transcendentals that sum to rationals
QUESTION [5 upvotes]: Is there any sequence $a_n$ of transcendental numbers, such that $ma_i\neq na_j$ for all integers m,n,i,j and any partial sum of $a_k$ is transcendental, but the total sum is rational ?

REPLY [4 votes]: Pick any rational $r$ you'd like for the sum, then choose rationals $r_1,r_2,\dots$ so that $\sum a_n=r$, 
where $a_n=r_n/\pi^n$.