## The Divergence of the Harmonic Series

The fact that the harmonic series, $1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + \cdots$, diverges has been known since the time of Nicole Oresme in the 14th century, but this fact is still somewhat surprising from a numerical standpoint.  After all, each successive term only adds a small amount to the total.  For example, if you add up the first 100 terms (i.e., $1 + 1/2 + \cdots + 1/100$) you only get a value of about 5.2.  Adding up the first 1000 terms only gives you a value of about 7.5.  Adding up the first 10,000 terms only gives you a value of about 9.8.

In fact, suppose you added up the first quintillion terms in the sum.  That’s $10^{18}$ terms, or 1,000,000,000,000,000,000 terms.  A quintillion is such a large number that if you traveled a quintillion centimeters you would be in the next solar system.  Yet adding up the first quintillion terms in the harmonic series would only give you a value of about 42.

The harmonic series grows very, very slowly.

Yet it does eventually get arbitrarily large.  To put that more precisely, no matter how large you posit a real number S there will always be some (probably quite large) integer N such that $1 + 1/2 + \cdots + 1/N > S$.

Let’s prove that the harmonic series diverges.  The proof is by contradiction.  First, suppose that the harmonic series converges to some number S, so that

$\displaystyle S = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \cdots.$

Let’s pair up the fractions on the right side, like so:

$\displaystyle S = \left(1 + \frac{1}{2}\right) + \left(\frac{1}{3} + \frac{1}{4}\right) + \left(\frac{1}{5} + \frac{1}{6}\right) + \left(\frac{1}{7} + \frac{1}{8}\right) + \cdots.$

Within each pair of parentheses, there’s a larger number and a smaller number.  If we replace the larger number in each pair with the smaller number, the value on the right must be smaller than it was before.  That leaves us

$\displaystyle S > \left(\frac{1}{2} + \frac{1}{2}\right) + \left(\frac{1}{4} + \frac{1}{4}\right) + \left(\frac{1}{6} + \frac{1}{6}\right) + \left(\frac{1}{8} + \frac{1}{8}\right) + \cdots \\ = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots \\ = S.$

This gives us $S > S$,  a logical contradiction.  Thus our assumption that the harmonic series converges must be false.  Therefore, the harmonic series diverges.

There are a lot of proofs that the harmonic series diverges.  For several of these, see Kifowit and Stamps [1].  (The proof I’ve given here is Proof 6 in their paper.)

Reference

1.  Steven J. Kifowit and Terra A. Stamps, “The Harmonic Series Diverges Again and Again,” AMATYC Review 27 (2), pp. 31-43, Spring 2006.