let give the sequence of integrals
J_n =∫_0 ^∞ x^n e^(−(x^2 /2)) dx
1) prove that J_n =(n−1)J_(n−2) ∀n≥2
2) calculate J_(2p) and J_(2p+1) by using factoriels.
3) prove that ∀n≥1 J_n ^2 ≤J_(n−1) . J_(n+1) .
4)prove that ((2^(2p) (p!)^2 )/((2p)!)) (1/(√(2p+1))) ≤J_0 ≤ ((2^(2p) (p!)^2 )/((2p)!)) (1/(√(2p)))
5) find a equivalent of ((2^(2p) (p!)^2 )/((2p)!)) (p→+∞)