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Saturday, November 2, 2013

A Simple Proof that the Harmonic Series Diverges

One usually encounters the harmonic series \[ \sum_{k=1}^{\infty} \frac{1}{k} \] as an example of a series that diverges for non-obvious reasons. With $H_n$ defined to be the partial sum $H_n:=\sum_{k=1}^n \frac{1}{k}$, a typical way to prove divergence is to observe that \begin{align*} H_{2n} &= H_n + \frac{1}{n+1} + \dots + \frac{1}{2n} \\ & \geq H_n + \underbrace{\frac{1}{2n} + \dots + \frac{1}{2n}}_{n \text{ terms}} \\ & = H_n + \frac{1}{2}. \end{align*} In particular, $H_{2^n} \geq 1 + \frac{n}{2}$, so the sequence of partial sums of the harmonic series has an unbounded subsequence, whence the series diverges.

The other day the author stumbled upon a simple proof of divergence. Almost certainly this is not new, due to the simplicity of the argument. Suppose for the sake of a contradiction that the harmonic series converges to a number $h$. Then \begin{align*} h &= 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots \\ & \geq 1 + \frac{1}{2} + \left( \frac{1}{4} + \frac{1}{4} \right) + \left( \frac{1}{6} + \frac{1}{6} \right) + \dots \\ &= \frac{1}{2} + h, \end{align*} an obvious contradiction.