Sum Of Partial Factorials
(December 23, 2006)
This
page shows some general formulas related to sum of partial factorials. There are many beautiful identities and
finite series that can be generated from these general formulas.
It is known that
or
and
so on, where 
.
Now we want to find the general formula of
or
for
all positive integers n.
We found a beautiful general formula that can be used to derive our desired formulas, namely
,where
.
We rewrite the above formula in another form as follows:
,
where
.
Consider some cases:
· Now let m = 2, we obtain a formula that can be proved by using the method of Mathematical Induction, namely
.
Example
Let n = 3 that both its sides are equal to 20.
.
· Let m = 3, we get the formula
.
Example
Let n = 2, both its sides are equal to 30.
.
· Let m = 4, we obtain
.
We see that it generates a formula for each value of positive m. Here are the lists of formulas that are true for all positive integers n.
·
or
.
·
or
.
·
or
.
·
or
.
·
or
.
· …
·
or
its summation notation is
,
where
.
We express it in terms of factorial form
and
its summation notation is
,
where m,
.
If we treat m as real x, we get the extended formula in terms of x as follows
and
,
where 
.
In addition, we get another interested formula that is based on the above results, namely
or
its expansion form is
,
where
Example
Let n = 4, we obtain a beautiful identity
for
all real x.
Thus, the above general formulas can generate many
identities and finite series that are true for all
. We also get many infinite series if the
values of x are in (-1, 1) as n move to infinity.
Here are the lists of identities that are true for all real x.
·
or
its summation notation is
.
·
or
.
·
or
.
· …
·
or
.
Sum
of Partial Factorial, from Series Math Study Web Resource.