Convergence of a series of exponentials

Question

Let $\{u_k\}_{k \geq 1}$ be a sequence of positive real numbers such that $u_k \to +\infty$. For $t>0$, consider the series $$\sum_{k=1}^{+\infty} e^{-u_k t}.$$ I am wondering if there always exists a $t>0$ such that this series converges (pointwise) ?

Answer 1

No I don't think so. For $k$ large enough set $u_{k} = \ln (ln k)$. Then \begin{equation*} \sum_{k = N}^{\infty}e^{-t(\ln{(\ln k})} = \sum_{k = N}^{\infty} \frac{1}{(\ln{k})^{t}} \end{equation*} which is divergent for all $t > 0$ by the Cauchy condensation test.