ABSTRACT
Since the addition of real numbers is a binary operation, that is, a map +: R× R → R, given an ordered pair (x, y) of real numbers we can add them to get x+y := +(x, y). It is the associativity of the addition that allows us to add a given finite ordered tuple (x1, . . . , xn). This is established in algebra books (especially in group theory). To “add” an infinite tuple, that is, to add a sequence, (xn) of real numbers, we need analysis to give a sensible meaning to “
∑∞ n=1 xn.” For
instance, if xn = (−1)n−1, and if we want to “add” them based on our earlier experience we may manipulate the terms and end up with “different” answers as follows.
