Chapter 14.5, Problem 57E

(a)

To determine

To calculate: The value of $T\left(1\right)$ and $T\left(2\right)$, if $T\left(x\right)={\displaystyle {\int }_0^{\infty }{t}^{x-1}{e}^{-t}dt}$.

(b)

To determine

To prove: For every positive integer $n$, $T\left(n+1\right)=nT\left(n\right)$.

(c)

To determine

To prove: The validity of the expression $T\left(n\right)=\left(n-1\right)!$ for every positive integer $n$.