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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

50 Identities of Power Summation


References

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


 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.

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