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)

Question:

For any positive integers m₁, m₂ and m₃,

Yes. 

 Other related series

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

50 Identities of Power Summation

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.

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