Assignment

For any integer $n>0,n!$ is defined as the product $nn-1n-2$dots$21.0!$ is defined to be $1.$ It is sometimes useful to have

a closed$-$form definition instead; for this purpose, an approximation can be used. R$.$W$.$ Gosper proposed the following such

approximation formula:

$n!$~~$n^n{e}^{-n}\sqrt[\phantom{2}]{(2n+\frac{1}{3})}$

Create a program that prompts the user to enter an integer $n,$ uses Gosper's formula to approximate $n!,$ and then displays the

result. The message displaying the result should look something like this:

$5!$ equals approximately $119\mathrm{.}97003$

Your program will be easier to debug if you use some intermediate values instead of trying to compute the result in a single

expression. If you are not getting the correct results, then you can compare the results of your intermediate values to what you get

when you do the calculations by hand. Use at least two intermediate variables$-$one for

$2n+\frac{1}{3}$

and one for

$\sqrt[\phantom{2}]{(2n+\frac{1}{3})}$

Display each of these intermediate values to simplify debugging. Be sure to use a named constant for PI$,$ and use the

approximation $3\mathrm{.}14159265.$ Test the program on nonnegative integers less than $8.$

Test Cases

$n,n!$

approximation

$8$

$40315\mathrm{.}903$