Repeated
Power Series
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. 
Example:
Let k=1 and n = 4, we see (1) is true, namely, 
, and
.
2. 
Example:
Let k=2 and n = 4, we see (2) is true, namely, 
and
.
3.
4. 
5. 
6. 
7.
or
Notice,
8.
9.
The above expression shows the
connection to Euler Constant, Sum of Power and Zeta Series.
Notice
that we have
Rewrite it
in term of summation notation
Multiplying
both sides of above equation by 
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)