Convergence of rapidly decaying series

Question

Let $0<\lambda_1<\lambda_2<\dots$ be an increasing sequence of positive real numbers with $\lim_{k\to\infty}\lambda_k=\infty$.

Let $a_1,a_2,\dots$ be a sequence such that for all $m>0$, $$a_k\leq O(\lambda_k)^{-m}.$$

Question: Does the series $\sum_k a_k$ converge absolutely?

Thoughts: Clearly if $\lambda_k=k$ are the natural numbers, then one large enough $m$ suffices. The question is whether the above "rapid decay" condition (with respect to an arbitrary sequence $\lambda_k$, tending to $\infty$) guarantee summability of $a_k$?

Answer 1

Note first that in the inequality $a_k\leq O(\lambda_k)^{-m}$ the $O$ must depend on $m$ since otherwise letting $m \to \infty$ gives $a_k=0$ as long as $O(\lambda_k)>1$ so the problem is trivial then.

The answer is then clearly no; consider the sequence $\lambda_k =\log k, k \ge 2$ and take $a_k=k^{-1/2}$ and for every $m \ge 1$ one has $k^{-1/2}=O_m((1/\log k)^m)$ while $\sum k^{-1/2}=\infty$