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 m_{1}, m_{2 }and m_{3},

**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.

**In-Text or Website Citation**

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

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