Sum Of Partial Factorials

Sum Of Partial Factorials
(December 23, 2006)
 
On this page, you'll find a commonly used general formula known as the sum of partial factorials. This formula is employed to derive numerous identities, each of which holds 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 there exist formulas of the form:
 
 or
 
or so on, where .
 
 
We want to find a general formula of the form:
 
 
.
 
 
or
 
 
 for all positive integers n.
 
The general formula (I) or (II) can be used to derive our desired formula. Let's consider some values of m:
 
,
which can be proved by using the method of mathematical induction.
 
It can easily be checked, for instance, n = 3, that both sides are equal to 20.
 
.
.
 
It can easily be checked, for instance, n = 2, that both sides are equal to 30.
 
                        .
 
.
 

Observing each positive integer m, we find that (I) yields a general formula in which it is true for all positive integers n.
 

,

or,

..

,

or,

.

,

or

.

or

..

or

.

General Formula

 

or it can be represented in summation notation as:

, where .
 
 
Furthermore, expressing it in the factorial form:
 
 
 
or in summation notation,
 
, where m, .
 
 
If we consider m as a real variable x, we get the extended formula in terms of x as follows:
 
 
 
 
or it can be represented in summation notation as:
 
            , where .
 
 
 
Moreover, an intriguing formula merges from the aforementioned results, denoted as:
 
 
or in expansion form,
 
 
, where .
 
 
Example
 
Let n = 4, we obtain a beautiful identity,
 
 
, for all real x.
 
 
The following are lists of identities for various values of n, all of which hold for real x.
 

 

 

or its product notation,

 

.

 
 

 

 

or in the product notation,

 

.

or

.

 

or it can be represented in product notation:  

.

 


Other finite series:

 


Number proceeds from unity. (Aristotle)