We know the elementary fact that if the partial sums $ \sum_{n\leq X} a_n $ are bounded, say by $ C$, then the series $ \sum_{n\geq 1} a_n n^{-s} $ converges for $s >0$.

My question then is, is there a simple upper bound in terms of $X$ and $C$ of the tail series $\sum_{n\geq X} a_n n^{-s} $ for a complex number $s $ with $\Re s>0$ ? The upper bound should tend to $0$ as $X\to \infty$.