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

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

References

In-Text or Website Citation
Tue N. Vu, Finite Series are Expressed in Terms of N-th Partial Sum of Hurwitz Zeta Function, 11/07/2009, from Series Math Study Resource.

 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.)