# Convergent series with divergent summands

Suppose we have a function $F:\mathbb{R}_{> 0} \rightarrow \mathbb{R}$ defined as the limit of the sum $$F(x) = \sum_{n=1}^{\infty} f(x,n).$$ This limit is well-defined for all positive $x$ and we know that $F\rightarrow 0$ as $x\rightarrow \infty$.

Is it possible for the summands $f$ to diverge as $x\rightarrow \infty$?

I would naively think that this is not possible, but maybe a subtle cancellation of divergencies could save the day here.

It is possible. Just take any non-constant analytical function that vanishes af infinity [e.g. $\exp(-x^2)$] and consider its Taylor-series.