 # A Brief Note on Nth Partial Sum of Harmonic Series

A Brief Note on Nth Partial Sum of Harmonic Series

(09/05/2006)

The purpose is to consider some series in connection with harmonic series and establish expressions in recurrence relation to harmonic number.

Some Series in Connection with Harmonic Series

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 - 1/22 ] by 1/[ (2k-1)2  + 1/22], its sum of alternating series converges, namely .

Expressions in Recurrence Relation to Harmonic Number

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)