Question

How do you find the sum of factorials \[1! + 2! + 3!.............. + n!\] ?

Answer 1

Hint: As we know factorial is the product of all positive integers less than or equal to a given positive integer and denoted by that number with an exclamation point and factorial of given number can be calculated using \[n! = n \times \left( {n - 1} \right) \times ..... \times 1\] and the sum of n factorials can be find using formula in terms of Euler’s Gamma function.

Complete step by step answer:
To find the sum of n factorial, we have a formula which computes the sum of factorials.
\[\sum\limits_{k = 0}^n {k! = \dfrac{{i\pi }}{e} + \dfrac{{{E_i}\left( 1 \right)}}{e} - \dfrac{{{{\left( { - 1} \right)}^n}\Gamma \left[ {n + 2} \right]\Gamma \left[ { - n - 1, - 1} \right]}}{e}} \]
Where,
 \[{E_i}\] is the Exponential Integral function
 \[\Gamma \left[ x \right]\] is the Euler Gamma Function whilst \[\Gamma \left[ {x,n} \right]\] is the upper incomplete Gamma Function.
\[e\] is Euler’s number and the value of e is 2.71828 and it is an irrational number.

Additional Information:
 Factorials are just products. An exclamation mark indicates the factorial. Factorial is a multiplication operation of natural numbers with all the natural numbers that are less than it, factorial is an important function, which is used to find how many ways things can be arranged or the ordered set of numbers.
The multiplication of all positive integers says “n”, that will be smaller than or equivalent to n is known as the factorial. The factorial of a positive integer is represented by the symbol “n!”.
Factorials are commonly encountered in the evaluation of permutations and combinations and in the coefficients of terms of binomial expansions. Factorials have been generalized to include non integral values. Factorial zero is defined as equal to 1.
Some of the examples are:
\[7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\]
\[1! = 1\]


Note: A factorial is defined as the product of the number with all its lowest value numbers. It is also defined as multiplying the descending series of numbers. The factorial formula is mostly used in permutations and combinations for probability calculation and to find out the factorial of number ‘n’ the formula is given as
\[n! = 1 \times 2 \times 3...... \times \left( {n - 1} \right) \times n\]
In this way we find the factorial of a given number.