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Sum Of Partial Factorials

Sum Of Partial Factorials
(December 23, 2006)
 
This page shows a general formula called sum of partial factorials that is used to generate many identities such that each of identity is true for all positive integers n.
 
General Formula
, where .
 
The general formula (I) can be rewritten in a product form as follows:
 
, where.
 

 
 
 
 
It is known that
 
 or
 
or so on, where .
 
 
Now we want to find a general formula which is of the form
 
 
 
 
or
 
 
 for all positive integers n.
 
 
Indeed, the general formula (I) or (II) is used to derive our desire. Let consider some values of m.
 
  • m = 2, we obtain a formula for all positive integers n, namely

 

, which can be proved by using the method of Mathematical induction.
 
It can easily be checked, for instance, n = 3, both sides are equal to 20.
 
.
 
 
  • m = 3, we get the formula
 
.
 
It can easily be checked, for instance, n = 2, both sides are equal to 30.
 
                        .
           
 
  • m = 4, we obtain

 

.
 
 
 

We see that for each positive m, (I) generates a general formula in which it is true for all positive integers n.
 
  • m = 1

 

 

or

 

.

 
 
  • m = 2

 

 

or

 

.

 
 
  • m = 3

 

 

or

 

.

 
 
  • m = 4

 

 

or

 

.

 
 
  • m = 5

 

 

or

 

.

 

 
 
 

General Formula

 

or its summation notation,

, where .
 
 
We express it in terms of factorial form
 
 
 
or its summation notation,
 
, where m, .
 
 
If we treat m as real x, we get the extended formula in terms of x as follows
 
 
 
 
or its summation notation,
 
            , where .
 
 
 
In addition, we get another interested formula that is based on the above results, namely
 
 
or its expansion form,
 
 
, where
 
 
Example
 
Let n = 4, we obtain a beautiful identity
 
 
for all real x.
 
 
Here are the lists of identities for different values of n. These identities are true for all real x.
 
  • n = 2

 

 

or its product notation,

 

.

 
 
  • n = 3

 

 

or its product notation,

 

.

 
 
  • n = 4

 

 

or

 

.

 
  •  

 

 
 

 

or its product notation,  

 

.

 


Other finite series:

 


Number proceeds from unity. (Aristotle)