TITLE: Is this a valid proof that the harmonic series diverges?
QUESTION [28 upvotes]: Is this a valid proof that the harmonic series diverges?

Assume the series converges to a value, S:

$$S=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...$$

Split the series into two, with alternating even and odd denominators. Since the original series converges, the component series will converge.

$$S_{EVEN}=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+...$$
$$S_{ODD}=1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+...$$
$$S=S_{EVEN}+S_{ODD}$$

Show that $S_{EVEN}=\frac{1}{2}S$

$$\frac{1}{2}S=\frac{1}{2}(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...)=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+...=S_{EVEN}$$

Show $S_{ODD}>S_{EVEN}$ because each odd term is greater than its corresponding even term:
$$1>\frac{1}{2}\qquad \frac{1}{3}>\frac{1}{4}\qquad \frac{1}{5}>\frac{1}{6}\qquad ...$$
Show $S_{ODD}=S_{EVEN}$
$$S_{ODD}=S-S_{EVEN}=S-\frac{1}{2}S=\frac{1}{2}S=S_{EVEN}$$
The contradiction implies that the original assumption of convergence is false:

$$S_{ODD}>S_{EVEN}$$
$$S_{ODD}=S_{EVEN}$$
$$\therefore S\ne 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...$$

REPLY [3 votes]: Rearrangement requires knowledge of absolute convergence. When I wrote up my proof and it was published 23 years ago, that was the comment added. Other than that, your proof is absolutely identical to mine. Here is the reference:
Michael W. Ecker, Divergence Of The Harmonic Series By Rearrangement, The College Mathematics Journal, May 1997, Vol. 28, No. 3, p. 209-210.
Several years later, Bernard August and Thomas Osler cited this and generalize this method in the May 2002 issue of The College Mathematics Journal, p. 233-234. If memory serves, they applied this to other p-series, but I don't have the issue in front of me.