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The title says it all, I'm curious if there are convergent infinite series whose terms are all real but which converge to a value that is not real. I want to phrase this as "are the reals closed under infinite summation" but I'm sure someone would on this site would fuss about that phrasing.

Two facts that I know make me wonder this:

First, not all series which are defined on rational numbers converge to a rational number. For example, the Basel problem wherein the sum of the squares of the reciprocal of natural numbers involves pi.

Second, I know that there are types of numbers on the number line that are not reals. The surreal numbers are the only case I know of, not to say there are not others.

These two facts together make me wonder if the reals can be used in this way (infinite summation) to define non-real numbers.

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  • $\begingroup$ It's worth mentioning there that 'on the number line' is a hazy and ill-defined concept. Most people would think of 'the number line' as being the real numbers, $\mathbb{R}$. The surreal, hyperreal, and complex number systems, to name a few, are number systems containing the real numbers as a subset, but few people would refer to them as 'the number line'. $\endgroup$
    – Albert Wood
    Commented Apr 20, 2022 at 2:43
  • $\begingroup$ @AlbertWood Fair enough! My implication with "on the number line" was meant to insinuate that it's not represent-able in multiple dimensions in a similar way that $\mathbb{C}$ is homeomorphic to $\mathbb{R}^2$. I don't have a firm enough grasp of these concepts to phrase it precisely, but by every description the surreals, for example, are "between" the reals, so "on the number line" seemed a reasonable way to hand wave that detail. If you know of a good way to phrase this please let me know. $\endgroup$
    – Michael New
    Commented Apr 22, 2022 at 15:14

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The answer is no (to your post's question, "yes" to your title's implied question), because the real numbers are a complete metric space. In other words, every summation that converges, converges to a real number.

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