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Does $\sum_{k\in\Bbb{R}^{[a,b]}}f(k)$ converge for specific non-constant $f:\Bbb{R}\rightarrow\Bbb{R}$ ?

I know it's possible to sum over sets like $\Bbb{Z}$ or $\Bbb{P}$ but I haven't really seen anything about summing over the reals even if we restrict the summation interval and the function we're using. Does anyone know if this would even work?

Harish Chandra Rajpoot
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hefe
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  • It is possible to define the sum of a function $f:S\to\mathbb{R}$ on an arbitrary set $S$ as follows: $$ \sum_{x \in S} f(x) = \lim_{\substack{F \uparrow S \ \text{$F$ finite}}} \sum_{x \in F} f(x) $$ This is a bit different from the ordinary summation for sequences, though. Indeed, $\sum_{x \in S} f(x)$ converges if and only if $\sum_{x \in S} |f(x)| $ converges. Consequently, conditional convergence, which critically hinges on the well-ordering structure of $\mathbb{N}$, does not occur for the sum $\sum_{x\in S}$. – Sangchul Lee Aug 05 '23 at 19:35

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