I am reading the excellent book How Not to Be Wrong by Jordan Ellenberg.
He points out the argument by Guido Grandi from 1703 that:
(1) Let $T = 1 - 1 + 1 - 1 + 1 - 1 + \dots$
(2) $-T = -1 + 1 - 1 + 1 - 1 + 1 - \dots$
(3) So, $-T = T - 1$ and $T = \frac{1}{2}$
Ellenberg writes:
Modern mathematicians would say that if we are to assign the Grandi series a value, it should be $\frac{1}{2}$, because, as it turns out, all interesting theories of infinite sums either give it a value of $\frac{1}{2}$ or decline, like Cauchy's theory, to give it a value at all.
How does one ever establish that all "interesting" theories of infinite sums give it a value of $\frac{1}{2}$ or decline to give it a value at all?
The author seems to be using the word "interesting" here as meaning that it has the following properties (where all sums runs $n=1$ to $\infty$ - I use the bare sum to emphasize that we are thinking of it merely as some function taking sequences to numbers):
If $s'_n$ is $s_n$ with a $0$ appended at the start, then $\sum s_n = \sum s'_n$.
$\sum a_n+b_n = \sum a_n + \sum b_n$ when both the latter exist.
If $s_n$ is zero for all $n>N$ then $\sum s_n = s_1+s_2+\ldots + s_N$.
We might also strengthen the last axiom to say that if $\sum_{n=1}^{\infty} s_n$ exists in the ordinary sense, then $\sum s_n$ agrees with it.
The point is that the above three axioms are sufficient to prove that the sum of $s_n=(-1)^{n+1}$ is $\frac{1}2$ if it exists. They are also very natural axioms to adopt to align with our intuition of a sum. So, the word "interesting" here really just means "acts like a sum" to exclude any operations $\sum$ that do something bizarre under ordinary circumstances.