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Let $\{a_{n}\}$ be a non-increasing sequence of positive numbers. if for some positive integers $l,p$ and $R>1$ we have $a_{(ln)^{p}}=O(R^{-n})$ as $n\to\infty$, what can we say about the behavior of $a_{n}$? tkx!!

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    Without further information, such as monotonicity, we cannot say anything further.2012-08-22

1 Answers 1

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If you consider $a_{i+1}\le a_i$ then

$f(n)=-\frac{1}{l}\sqrt[p]{n}$

$a_n=O(R^{f(n)})$

DETAILS :

consider $m$ such that $(lm)^p\le n \lt l(m+1)^p$, then $m=\lfloor\,f(n)\rfloor$

As $a_{(ln)^p}=O(R^{-n})$, there exists $k>0$ such that $a_{(ln)^p}\le k(R^{-n})$

So for any $n$ (and $R\ge 1$), $a_n

and if $(R<1)$

$a_n

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    Could you give some details ? Please?2012-08-23