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Sum Of Partial Factorials

Submitted by admin on Tue, 03/17/2009 - 3:23pm
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

  • and so on.
 
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
  • n = 2:

 
  • n = 3:

 

 

  • n = 4

 

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: