**A Brief Note on Nth Partial Sum of Harmonic Series**

(09/05/2006)

The harmonic series is defined as the sum of 1, 1/2, 1/3, …, and it is written in expanded form with nth partial summation notation of harmonic series as follows:

Its sum diverges to infinity as n tends to infinity, namely

.

The alternating harmonic is defined as the sum of 1, -1/2, 1/3, -1/4, … . Its sum converges to ln (2), namely

.

Rewrite the alternating series in the form of even and odd harmonic series as follows:

.

Therefore, the odd harmonic series also diverges to infinity.

.

The odd harmonic series is rewritten in another form as shown in the following steps:

Therefore,

.

Now we see that the harmonic series is transformed into the form of

.

However, if the k term from the nominator of the above expression is removed, we then obtain

.

If we replace 1/[ (2k-1)^{2} - 1/2^{2 }] by 1/[ (2k-1)^{2} + 1/2^{2}], its sum of alternating series converges, namely

.

Define

is called harmonic number. (Notice for different notation in Wiki.)

We then obtain the following recurrence relation:

However, if we do the recursive substitution for , it gives a simple relationship between , namely

It is also easy to see that

.

Thus, we can state harmonic number as follows:

(09/05/2006)