grandes-ecoles 2018 QIV.1

grandes-ecoles · France · x-ens-maths__pc Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula

For $n \in \mathbb{N}$, we set $$I_{n} = \int_{0}^{+\infty} x^{n} e^{-x}\, dx.$$ Determine by induction $I_{n}$ for all $n \in \mathbb{N}$.

For $n \in \mathbb{N}$, we set
$$I_{n} = \int_{0}^{+\infty} x^{n} e^{-x}\, dx.$$
Determine by induction $I_{n}$ for all $n \in \mathbb{N}$.