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**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:

- 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, that both sides are equal to 20.

.

- m = 3, we get the formula:

.

It can easily be checked, for instance, n = 2, that both sides are equal to 30.

.

- m = 4, we obtain

.

Observing each positive integer m, we find that (I) yields 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 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.

- n = 2

or its product notation,

.

- n = 3

or in the product notation,

.

- n = 4

or

.

- …

or it can be represented in product notation:

.

**Other finite series:**

- Finite Alternative Odd Power Series
- Finite Power Series
- Finite Odd Power Series
- Repeated Power Series
- Some Finite Series Found in Closed-Form

(Aristotle)Number proceeds from unity. |

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