Is it always possible to find a $k \in \mathbb{N}$ such that for a sequence $a_n$, where $\lim\limits_{n \rightarrow \infty} a_n=0$, the sum $$\sum\limits_{n=0}^{\infty}a_n^k < \infty $$ I suspect there is some pathological counterexample, but I have not yet found it. Any ideas?
2026-05-05 10:48:29.1777978109
Existence of $k$ such that for $\lim\limits_{n \rightarrow \infty} a_n=0$, $\sum\limits_{n=0}^{\infty}a_n^k < \infty$
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