Finite Series in Connection with Apéry, Pi Constant

The n-th partial sums below are true for each positive integer n.

•   (I)

 

• (II)

 

The notationsandin (I) and (II) represent the special values of a new generic formula, which we define as an extensive notation from Hurwitz zeta function [1*] for n-th partial sum as follows: 

,

where s and a are complex variables with Re(s) > 1 and Re(a) > 0.

 

When n tends to infinity, the extensive notation [2*] is expressed as

.

 

Special Case

• n = ∞:

As n approaches to infinity, both series (I) and (II) converge to the following values:

,

where is Apéry's constant.

 

.

 

Example

• If we put n = 2, both sides of (II) are equal to 29/31752, namely

 .

 

 (November 26, 2009 - Happy Thanksgiving)

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Question:

For any positive integers m₁, m₂ and m₃,

 

Yes. 

 

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 Other related series

Finite Series are Expressed in Terms of N-th Partial Sum of Hurwitz Zeta Function

50 Identities of Power Summation

 

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References

[1*] Hurwitz function, en.wikipedia.org/wiki/Hurwitz_zeta_function, from Wikipedia resource.

[2*] The purpose of our website is to show a beauty of series.  We introduce a new extension of the notation of Hurwitz zeta function for n-th partial sum because there exist such series (I) and (II).  We share our results on the internet.  The acceptance of this notation is the work of other men.

 

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 God speaks to us in many ways. Math is one of them. (T.V.)