Let $c\in(0,1)$, $m\geq 1$ be positive integer and $\{a_{n}\}$ a decreasing sequence of positive real numbers. Suppose that $$a_{n^{m}}\leq K c^{n}n^{-m/2}, \forall n\in\mathbb{N}, $$for some $K>0$. How to describe the behavior of the whole sequence $\{a_{n}\}$?
Asymptotic behavior of a sequence based on a subsequence.
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sequences-and-series
asymptotics