TITLE: Why do we ask for *absolute* convergence of a series to define the mean of a discrete random variable?
QUESTION [28 upvotes]: If $X$ is a discrete random variable that can take the values $x_1, x_2, \dots $ and with probability mass function $f_X$, then we define
its mean by the number $$\sum x_i f_X(x_i) $$ (1)
when the series above is absolutely convergent.

That's the definition of mean value of a discrete r.v. I've encountered in my books (Introduction to the Theory of Statistics by Mood A., Probability and Statistics by DeGroot M.).
I know that if a series is absolute convergent then it is convergent, but why do we need to ask for the series (1) to converge absolutely, instead of just asking it to converge? I'm taking my introductory courses of probabilty and so far I haven't found a situation that forces us restrict ourselves this way.
Any comments about the subject are appreciated.

REPLY [2 votes]: There is a distinction to be made in math between features that are an essential part of a mathematical object, versus features that are used in a more arbitrary manner simply as labels so that we can discuss the objects.
For instance, if we're encoding demographic data, and one of the variables is race, to encode that data as an integer requires assigning numbers to the different races. What number a race gets is not an essential feature of the race, but simply an arbitrary number given to keep track of it. It would be rather weird if our calculation of the average value of some metric over the different races gave a different answer depending on how we labeled the races.
Similarly, the index that we assign different $x$ values is not an essential part of the data. Even if there's an "obvious" order, it's not the only ordering, and we would want our definition to not depend on picking the right order, or there being an obvious ordering. Consider the set of rational numbers. This set is countable, so it's possible to come up with a labeling of the rational numbers by integers, but there are many different ways of going about this. If two people used different methods for labeling the rational numbers and got different numbers for the mean, that would be a problem.
The fact that the formula has $\Sigma$ rather than $\Sigma_{i=0}^{\infty}$ emphasizes this: the mean is an attribute of the  events and their probabilities, not of any particular indexing. We can define the $\Sigma$ operator without any reference to an indexing. The sum over a finite set can easily be defined, and for an infinite set $X$, we can define the limit of its sum as being the number $L$ such that for any positive $\epsilon$, $X$ can be split into a finite set $H$ and infinite set $T$ such that the sum over $H$ plus the sum over any finite subset of $T$ is within $\epsilon$ of $L$. i.e.
$\Sigma X = L$ if $\forall \epsilon >0,\exists H,T:H \cup T = X, H \cap T= \emptyset, |H|<\infty, \forall S \subset T, |S| < \infty \rightarrow |\Sigma X + \Sigma S-L|<\epsilon$
This definition (unless I've messed up somewhere) is equivalent to "uniform convergence"; the term "uniform" is a qualifier we have to add when defining sums with respect to  an indexing to make that sum independent of the indexing.