The Harmonic Series

(A downloadable pdf version of this article will be made available soon)

Well, most of the people who have had a brush with mathematics are at least a bit familiar with the standard progressions. The Arithmetic Progression where the difference between adjacent terms is constant, the Geometric Progression where the ratio of the adjacent terms is constant and the harmonic Progression in which the difference between the reciprocals of the adjacent terms is constant.

Here I wish to point out certain queer results about this amazingly simple series. The theorems themselves have been known for long and are exciting as well as simple enough to understand and appreciate.

Let us first define the Harmonic Series :

The Harmonic Series is defined as

Note that the terms of the series are in Harmonic Progression ( hence the name).

Result : Observing the terms of the series we see that each term is smaller than the preceding one. A reasonable prediction would be that the harmonic series converges to a number( real ). However this is one of the first suprising result about this series. It can be easily proved that the Harmonic Series diverges. We record this beautiful result as a theorem.

Theorem : The Harmonic Series Diverges.

Proof: For the definition of convergence check The Convergence Page.

We define as the nth harmonic number.

For a series to converge the sequence formed by the nth partial sum must converge.

We show that the nth partial sum exceed every conceivable positive number.

Let K be a positive integer. We show that for sufficiently high value of n exceeds K.

The trick is to recognise that

Similarly,

So we see that starting with ½ and grouping terms in the nth harmonic number in sets of elements we obtain that each of these sets exceeds ½ .

Since the set ends at

we choose the .

Now from the argument given above it follows that the

exceeds K as required.The proof is complete here.

For someone who doesn't understand how does divergence follow from here:

Note that we have shown that for any arbitrary K there exists a value of n such that nth Harmonic number exceeds K. This implies that appropriately choosing n for any K we can show that the Harmonic series would conquer it. Thus Harmonic number must exceed every possible positive integer( with some adjustment even all reals). This is indeed the intuitive meaning of "tending to infinity" or "divergence". More on this in the specific section on Cauchy's definition of Limit and Convergence.