TITLE: Does there exist a sequence of real numbers $\{a_n\}$ such that $\sum_na_n^k$ converges for $k=1$ but diverges for every other odd positive integer?
QUESTION [11 upvotes]: Does there exist a sequence of real numbers $\{a_n\}$ such that $\sum_na_n^k$ converges for $k=1$ but diverges for every other odd positive integer?

REPLY [7 votes]: Yes. In fact, a stronger fact holds: 

For any set $C$ (finite or infinite) of odd positive integers, there is a sequence of real numbers $\{a_n\}$ such that for $k$ odd, 
   $$ \sum_n a_n^k $$
  converges iff $k\in C$.

Whether this is possible was asked by Polya as problem 4142 in the American Mathematical Monthly. It was solved by N.J. Fine, the solution appeared in 1946 (pp 283-284), and can be found here.