Mathematics

Problem

For a non-negative integer

n

, define

J_n=\int_{0}^{\infty}x^{n}e^{-x}\,dx.

(You may assume that each integral converges.) a. Show that

J_n=n\,J_{n-1}

for every integer

n\ge1

. State each step of your reasoning explicitly. b. Compute

J_0

exactly. c. Combine parts (a) and (b) to obtain a closed-form expression for

J_n

as a function of

n

only. Simplify your answer as far as possible. d. Determine the numerical value of

J_6

. Express your answer as an integer. e. Discuss the behavior of the sequence

\displaystyle\frac{J_n}{n!}

n\to\infty

. Provide a formal justification for your conclusion.