I'm currently reviewing some mathematics for a thermal physics course. My textbook claims that the factorial integral $$ n! = \int_0^\infty x^n\text{e}^{-x}\text{d}x$$ can be proved via induction. My current work:
- n = 0: $$\int_0^{\infty}x^0\text{e}^{-x}\text{d}x = \int_0^{\infty}e^{-x}\text{d}x = 1$$ which is 0!.
- Assuming that the case $n = k$ is true $$k! = \int_0^{\infty}x^k\text{e}^{-x}\text{d}x$$ we now wish to show that the $n = k+1$ case follows. The textbook then gives a hint to integrate the $k+1$ case by parts. First rewriting the integral $$(k+1)! = \int_0^{\infty}x^{k+1}\text{e}^{-x}\text{d}x = -\int_0^{\infty}x^{k+1}\frac{\text{d}}{\text{d}x}(e^{-x})\text{d}x$$ and then swapping over the derivative $$ -\int_0^{\infty}x^{k+1}\frac{\text{d}}{\text{d}x}(e^{-x})\text{d}x = \int_0^{\infty}\frac{\text{d}}{\text{d}x}(x^{k+1})\text{e}^{-x}\text{d}x + \left[x^{k+1}\text{e}^{-x}\right]_0^{\infty} $$ The proof would be complete if the boundary term (second term on the right-hand side) comes out to be 0; however, I'm getting $-x$. What am I doing wrong? Thanks in advance for any help.
$x^k e^{-x} \to 0$ as $ x \to \infty$