Does every infinite summation with real terms converge to a real number? 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.

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.
