Given a sequence $a_n$ such that $\sum_{n\ge1} \dfrac{a_n}{n^s}$ is convergent for $s>0$. Given that $\sum_{n\ge1} \frac{a_n}{n} < 1$, is it be possible to impose some sort of an upperbound for $\sum_{n\ge1} \frac{a_n}{n^s}$ for $1>s>k>0$ for some fixed $k$?
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Given an integer $m$, define a sequence $a_n$ to be $m$ at the $m$th place and $0$ otherwise. Then $$\sum_{n\ge 1} {a_n\over n^s}=m^{1s}$$ As any upper bound must work for all $m$, we see that no upper bound is possible for $s<1$. 

