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.

 

 

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