Question

The value of \[\dfrac{1}{2}\]! \[ + \]\[\dfrac{2}{3}\]! \[ + \].........\[ + \]\[\dfrac{{999}}{{1000}}\]! is

Answer 1

Hint: In this question, we have to find the value of a given series based on factorial. We will proceed by writing the numerator of each terms of the given series as the difference of its denominator minus one( i.e. \[\dfrac{1}{{2!}} = \dfrac{{(2 - 1)}}{{2!}}\] ). Now we will distribute the denominator over numerator
(i.e. \[\dfrac{{(2 - 1)}}{{2!}} = \dfrac{2}{{2!}} - \dfrac{1}{{2!}}\] ) and then cancel the same terms to get the resultant solution of the series.

Complete answer:
This question is based on sequence and series of factors. A factorial in mathematics is a function that multiplies a number by every number below. it is denoted by the symbol ! . For example , \[5! = 5 \times 4 \times 3 \times 2 \times 1\] , \[6! = 6 \times 5 \times 4 \times 2 \times 1\] and so on.
Consider the given question,
We have to find the value of \[\dfrac{1}{2}\] factorial\[ + \]\[\dfrac{2}{3}\] factorial \[ + \].........\[ + \]\[\dfrac{{999}}{{1000}}\] factorial
Let , \[x = \dfrac{1}{{2!}} + \dfrac{2}{{3!}} + .............. + \dfrac{{999}}{{1000!}}\] , where symbol \[!\] denote factorial in mathematics .
The above expression can also be written as ,
\[ \Rightarrow x = \dfrac{{(2 - 1)}}{{2!}} + \dfrac{{(3 - 1)}}{{3!}} + .............. + \dfrac{{(1000 - 1)}}{{1000!}}\]
distributing the denominator over numerator , we have
\[ \Rightarrow x = \dfrac{2}{{2!}} - \dfrac{1}{{2!}} + \dfrac{3}{{3!}} - \dfrac{1}{{3!}} + .............. + \dfrac{{1000}}{{1000!}} - \dfrac{1}{{1000!}}\]
Cancelling the common term from numerator and denominator we have,
\[ \Rightarrow x = \dfrac{1}{{1!}} - \dfrac{1}{{2!}} + \dfrac{1}{{2!}} - \dfrac{1}{{3!}} + .............. + \dfrac{1}{{999!}} - \dfrac{1}{{1000!}}\]
Now we can see that in the above series, each term gets cancelled except the first and last term. Hence, we have
\[ \Rightarrow x = \dfrac{1}{{1!}} - \dfrac{1}{{1000!}}\]
On simplifying, we get
\[ \Rightarrow x = \dfrac{{1000! - 1}}{{1000!}}\]
Hence, the value of the given series \[\dfrac{1}{2}\] factorial\[ + \]\[\dfrac{2}{3}\]factorial \[ + \].........\[ + \]\[\dfrac{{999}}{{1000}}\] factorial is \[\dfrac{{1000! - 1}}{{1000!}}\].

Note:
Factorial is a function that multiplies a number by every number less than itself. For example: \[n! = n(n - 1)(n - 2)(n - 3).......(3)(2)(1)\] where \[n\] is any integer greater than \[1\].
We define zero factorial equal to one. (i.e. \[0! = 1\] ).
Factorial of Negative integers are not defined.