The integral function $(n)={\displaystyle \underset{0}{\overset{}{}}}{t}^{n-1}{e}^{-t}dt$ is defined for all real numbers $n$ such that $n1$

$($a$)$ Show that the integral defining $(n)$ converges whenever $n1.$

$($Hint: Explain why there exists a positive number $M$ such that $e^t{t}^{n+1}$ for all $tM.$ Then split up the integral for $(n)$ at $t=M$ and use the Comparison Test on the integral from $M$ to infinity. What is your comparison integral?$)$

$($b$)$ Using integration by parts, show that $(n+1)=n*(n).$

$($c$)$ Show that for $(n+1)=n!,$ for every integer $n1.)$ Conclude that the function $(n)$ is an extension of the factorial function $n!$ to the case where $n$ is not an integer.