Application. For all $n \in \mathbb { N } ^ { * }$, we denote $\Gamma ( n ) = \int _ { 0 } ^ { + \infty } x ^ { n - 1 } e ^ { - x } d x$. (a) Calculate $\Gamma ( n )$ for all $n \in \mathbb { N } ^ { * }$. One will use induction. (b) Deduce the following asymptotic equivalent $$n ! \underset { n \rightarrow + \infty } { \sim } \sqrt { 2 \pi } n ^ { n + 1 / 2 } e ^ { - n }$$ Hint. First rewrite $\Gamma ( n + 1 )$ in the form $$\Gamma ( n + 1 ) = n ^ { n + 1 } \int _ { 0 } ^ { + \infty } e ^ { - n ( x - \ln x ) } d x$$
Application. For all $n \in \mathbb { N } ^ { * }$, we denote $\Gamma ( n ) = \int _ { 0 } ^ { + \infty } x ^ { n - 1 } e ^ { - x } d x$.\\
(a) Calculate $\Gamma ( n )$ for all $n \in \mathbb { N } ^ { * }$. One will use induction.\\
(b) Deduce the following asymptotic equivalent
$$n ! \underset { n \rightarrow + \infty } { \sim } \sqrt { 2 \pi } n ^ { n + 1 / 2 } e ^ { - n }$$
Hint. First rewrite $\Gamma ( n + 1 )$ in the form
$$\Gamma ( n + 1 ) = n ^ { n + 1 } \int _ { 0 } ^ { + \infty } e ^ { - n ( x - \ln x ) } d x$$