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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.

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

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  • $\begingroup$ That is astonishingly simple...Thanks $\endgroup$
    – Hrodelbert
    Aug 14 '15 at 10:51

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