grandes-ecoles 2018 Q9

grandes-ecoles · France · centrale-maths1__pc Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula

We denote $M_{p} = \int_{-\infty}^{+\infty} x^{2p} \exp\left(-x^{2}\right) \mathrm{d}x$. For $p$ a natural integer, give a relation between $M_{p+1}$ and $M_{p}$ and deduce that, for all $p \in \mathbb{N}$, $$M_{p} = \frac{\sqrt{\pi}(2p)!}{2^{2p} p!}$$

We denote $M_{p} = \int_{-\infty}^{+\infty} x^{2p} \exp\left(-x^{2}\right) \mathrm{d}x$. For $p$ a natural integer, give a relation between $M_{p+1}$ and $M_{p}$ and deduce that, for all $p \in \mathbb{N}$,
$$M_{p} = \frac{\sqrt{\pi}(2p)!}{2^{2p} p!}$$

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