Up until now, I only knew a couple of standard proofs that the harmonic series diverges: 1) 1 + 1/2 + (1/3 + 1/4) + (1/5 + ... + 1/8) + ... > 1/2 + 1/2 + 1/2 + 1/2 + ... 2) integral(1/x) = ln x I think that's about it ... But then I saw this very very cool proof idea in what was otherwise a mostly weak book of philosophy of mathematics ( http://www.amazon.com/How-Mathematicians-Think-Contradiction-Mathematics/dp/... if you are interested ) and I think it could be said something like this: Suppose it converges. Then let H = 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... so 1/2 H = 1/2 + 1/4 + 1/6 + 1/8 + ... and, subtracting, H - 1/2 H = 1 + 1/3 + 1/5 + ... Finally comparing term-for-term, it's clear that 1/2 H > 1/2 H, contradiction, QED. Does anyone have an early citation for this proof, or know who might have thought it up? Byers doesn't give any citation for the idea in his book. Are there any major holes in the proof that need to be filled, or is this basically a good outline? (Since we assume convergence to get the contradiction, I think the manipulations in the middle are OK, right?) Thanks, --Joshua Zucker