The Harmonic Series

I've been daydreaming about how I would teach an introduction to calculus in just two hours. It's hard because when I start thinking about calculus, which I generally take for granted, I begin to see holes and gaps in my understanding. (What precisely is an "infinitesimal" differential element supposed to mean, anyway?)

Just now, instead of another hole, I found this nice smooth pebble, my first for this blog. It's a simple proof that the harmonic series diverges. The harmonic series is this one:




The sum "goes to infinity", meaning that if you take any number at all, after enough terms the harmonic series grows bigger than it. The proof proceeds via reductio ad absurdum.

Assume the limit is some number S. Then break the sum into two smaller sums, one of the odd terms and one of the even terms, like so:










All the terms from the original sum are there in one of those two smaller guys. Also, s₁,the sum of odd terms, must be greater because each individual element is greater (i.e. 1> 1/2, 1/3 > 1/4, etc.)
So algebraically:





Now for the trick. Just multiply s₂ by 2.







If s₂ is half of S, then s₁ must be the other half.




Wait! s₁ is supposed to be bigger, but now they're equal to the same thing. To check, plug (8) and (9) into (6), and you're left with:




Which is a contradiction. Therefore, the hypothesis is wrong, and the sum is not a finite number. It is positive, since each of its terms is, and so the sum simply gets bigger and bigger without bound.

After a quick googling, it turns out my proof is number 8 on a long list (continued here) of proofs, but that no one found it until 1979!