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PART-1
If d ,
t are finite integers such that (t
³ d) and n is
an integer going to
Infinity. and
i stands for 
then we may define p by the following relation.
Where ( ln ) stands for Natural
logarithm.
The real part converges to p , while the coefficient of the imaginary part converges to zero.
Please click here to see the
related Mathematica program
This theorem has many interesting
properties . If (t - d = 0) that is d=t then
it is enough to only give n=1
value , p will
converge to real p .
But
as (t - d) increases we need more and
more n , (up to infinity at last)
to converge the
p for a
certain accuracy to hold .
PART-2
If d ,
t are finite integers such that (t
³ d) and n is
an integer going to
Infinity. and
i stands for then we may define p by the following relation.
Where ( ln ) stands for Natural logarithm.
The real part converges to p , while the coefficient of the imaginary part converges to zero.
Please click here to see the
related Mathematica program
This theorem has many interesting
properties . If (t - d = 0) that is d=t then
it is enough to only give n=1
value , p will
converge to real p .
But
as (t - d) increases we need more and
more n , (up to infinity at last)
to converge the
p for a
certain accuracy to hold .
PART-3
If d ,
t are finite integers such that (t
³ d) and n is
an integer going to
Infinity. and
i stands for then we may define p by the following relation.
Where ( ln ) stands for Natural logarithm.
The real part converges to p , while the coefficient of the imaginary part converges to zero.
Please click here to see the
related Mathematica program
This theorem has many interesting
properties . If (t - d = 0) that is d=t then
it is enough to only give n=1
value , p will
converge to real p .
But
as (t - d) increases we need more and
more n , (up to infinity at last)
to converge the
p for a
certain accuracy to hold .
It is obvious that
anybody may play with the summation limits , since it doesn't make any
difference as one of the variables
in the summation
limits goes to infinity. But before doing this we strongly advice to think
carefully. Because there is a
reason for these
strange limits; as you approximate one or two of the variables to one or
zero , you may lose some of the
correct answers
on singular points ; Such as (d=1 , t=1 , n=1 ) . You
will discover some additional singular points as
you
exprience more ond more on these functions. After all , the success of a
theorem or a formula depends on the output
that it
produces on any odd or singular input that is given to it.