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Repeated Power Sum

Repeated Power Sum
(June 27, 2005)
 
The followings are repeated power sum formulas that are true for all positive integers. The readers can prove them by using principle of Mathematical Induction.   (The way of how the Conjecture of these formulas is established is a different story.   It connects to the work of applying zeta(2) in the purpose of finding the exact formulas for calculating the integrated intensity, power and energy of white-light rays in term of time by letting white light beam with initial intensity * travels in closed-paths so that its initial rays are returned back to the same its initial direction and plane as where it begins. This technique is called an integrated white light method that can be used to build a powerful intensity light source from a small intensity light source. We are still working on it, and will publish this article in the Blog section when done.)
 
We introduce here the new form of some formulas that have been found with coming up a Conjecture in the following:
 
 Assuming that is defined as follows
, where are positive integers.
 
We define two expressions
 
and
,
where is represented for repeated power series.
 
We prove that
or
 
 
Based on our initial assumption and definition, we obtain the repeated power series formulae of for k = 1 to 7 as follows:
 
1.       or or
 
 .
 
Example:
Let k=1 and n = 4, we see both sides of (1) are equal to 20. Indeed, the left hand of (1) is , and the right hand of (1) is
.
     
 
2.       or
 
.
 
Example:
Let k=2 and n = 4, we see both sides of (2) are equal to 50.  Indeed, the left hand of (2) is
, and the right hand of (2) is
            .
 
 
 
3.       or
 
.
 
 
4.       or
 
.
 
 
5.       or
 
.
 
 
6.       or
 
.
 
 
7.       or
 
.
 
Notice,
.
 
 
 
8.             .
 
 
9.       .
 
The expression (9) shows the connection to Euler Constant, Sum of Power and Zeta Series.
 
 
Notice that we have
.
 
Rewrite it in term of summation symbol
 (**)
 
Multiplying both sides of (**) by , we have
 
           .
Therefore,
 
        .
 
We thus obtain a remarkable formula that shows the connection to Euler Constant, Sum of Power, and Zeta function.
,
 
where
 
Or , where A is Glaisher’s constant = 1.28243…
 
 
(June 27, 2005)
 
 

In mathematics when we face a critical problem we cannot solve it with the obvious proof, we understand that the current math tools have failed to reach beyond it. This implies that we need to develop another mathematic tool to solve for it, and this process would  take centuries. (T.V.)
 

Other finite series: