Submitted by admin on Mon, 09/28/2009 - 10:46pm
A Brief Note on Nth Partial Sum of Harmonic Series
(09/05/2006)
The purpose is to examine certain series related to the harmonic series and establish expressions involving recurrence relations to harmonic numbers.
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


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)
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