Sum Of Partial Factorials
Sum Of Partial Factorials
(December 23, 2006)
We present a general formula called as the 'sum of partial factorials'.
General Formula
for m and n ∈ N. Formula (I) holds true for all positive integers m and n. The term in the formula can be expressed in expanded form as:
for m and n ∈ N.
It is known that there exist finite series of the form:
-

-

-
and so on
for
.
We endeavor to identify other formulas with a similar pattern to the one described above by studying the characteristics of the subsequent finite series through the method of induction:
or express in the sum notation

or

or

or

The above pattern can be generalized as:
or
where m and n are positive integers.
Furthermore, it can be expressed in the factorial form:
or
for m and
.
We discover that the sum still holds when we treat m as a real variable x,
or
for x ∈ R and n ∈ N.
Finally, we rewrite it in the summation and production notations as:
for x ∈ R and n ∈ N.
Examine various values of n



Later, we note that any forms of the finite series can be obtained by applying the standard formulas for sums of integers powers.
Other finite series: