Let $a_n > 1, n>0$ be an increasing unbounded sequence of real numbers. Is it true that $\sum_n e^{-a_nz}$ converges for all $z>0$? If not, then is it at least true that $\sum_n e^{-a_nz}(\log a_n)/a_n$ converges?
Standard tests (like the ratio test) do not help here, I think. Any help appreciated!