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?
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?
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.