**Brief Note** - Introduction a generalization form of n-partial sum of Hurwitz Zeta Function.

The n-th partial sums of the expressions (I) and (II) below are found in closed-form for each positive integer n and real x ≠ -k and x ≠ -(k+1).

.

.

**Special values**

- When n tends to infinity, (I) is reduced to a simple form

.

Let x = 0,

.

- Similarly, (II) gives

. (Correct - 11/17/2009)

At x = 1,

.

**Example **

- It is easy to verify that for n = 2, both sides of (I) are equal for all x ≠ - 1, -2, and -3, namely

.

**A Generalization Form of n-th Partial Sum of Hurwitz Zeta Function**

Recall that Hurwitz Zeta Function [1] is defined for complex arguments s and *a* by

, where Re(s) > 1 and Re(a) > 0.

We now define a new generalization form of n-th partial sum of Hurwitz zeta function above as follows:

.

Based on this new define, (I) and (II) are then rewritten in terms of n-th partial sum of Hurwitz zeta function for Re[*a*] = x, R[s] = 2, and each positive integer n as shown in the following:

(III) .

(IV) .

The identities (III) and (IV) are also true for complex number *a* by replacing x = *a.*

(November 07, 2009)

(Update formula syntax and definition - December 05, 2009)

**Other related series**

50 Identities of Power Summation

**References**

[1] http://en.wikipedia.org/wiki/Hurwitz_zeta_function

**In-Text or Website Citation**

Mathematicians might have only acknowledged the beauty of mathematical formulas silently after understanding them. The exquisiteness on human faces is an ephemeral beauty but the charming manifestation of mathematics is a beauty in all eras. (T.V.) |

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