Given an infinite sequence of positive numbers, one way to define their infinite sum is as the least upper bound (if any) of the sums of finitely many of the numbers:

(Of course, the condition that the numbers be positive is crucial, here.) The cool thing about this definition is, unlike the usual epsilon-delta definition, this one doesn’t actually require that the set of positive numbers be countable. This allows us to define the infinite sum of uncountably many positive numbers.

This definition doesn’t actually give us anything new, though. See, if S really is uncountable, then it turns out the sum of S automatically diverges. The proof of this fact is a very cool use of the higher-cardinality Pigeonhole Principle. The finite pigeonhole principle says: If you have more pigeons than pigeonholes, then some pigeonhole is going to need to have more than one pigeon put in it. The higher-cardinality version says that if you have infinitely many pigeons, and the cardinality of your pigeons exceeds the cardinality of your pigeonholes, then some hole is going to need infinitely many pigeons.

  • Theorem: If S is any uncountable set of positive numbers, then the sum of S diverges.
  • Proof:
    • Separate the positive numbers into the intervals (1,infinity), (1/2,1], (1/3,1/2], (1/4,1/3], (1/5,1/4], and so on.
    • I’ve separated the positive number line into countably many intervals.
    • By assumption, S is uncountable. Consider numbers in S to be pigeons, and intervals in the partition to be pigeonholes. There are uncountably many “pigeons” and only countably many “holes”. Thus, some “hole” must get infinitely many “pigeons”.
    • In other words, some interval (1/k,1/(k-1)] (or (1,infinity)) contains infinitely many elements of S.
    • Everything between 1/k and 1/(k-1) is at least as big as 1/k. So S contains infinitely many things which are all at least as big as 1/k. (And this is still true, with k=1, if the interval with infinitely many elements is (1,infinity))
    • It follows that the sum of S diverges. If M is any real number, however huge, we can find an integer n so big that n/k>M. Some finite subset of S contains n things, all at least as big as 1/k. The sum of this finite set is at least n/k>M. So no number at all is big enough to bound the sum of S, and the sum diverges.

This explains why authors don’t use the supremum-of-finite-sums definition for infinite series, and use epsilon-delta instead. The former adds nothing new, while the latter allows negative numbers as well as positive.

FURTHER READING

Covering the Rationals
Conic Sections in Real Life
Konig’s Lemma