- #1

Sigma057

- 37

- 1

## Homework Statement

I've begun going through Boas'

*Math Methods in the Physical Sciences*and am stuck on problem 1.15.25. The problem is to evaluate

## \lim_{x\to \infty } x^n e^{-x} ##

By using the Maclaurin expansion for ##e^{x}##.

## Homework Equations

We know the Maclaurin expansion for the exponential function to be

## 1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\text{...}##

We are also given the hint to divide the numerator and denominator by ##x^n## before taking the limit.

## The Attempt at a Solution

Taking the hint I proceed

##

e^{-x} x^n=\frac{x^n}{e^x}=\frac{1}{\frac{e^x}{x^n}}=\frac{1}{\frac{1}{x^n}\left(1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\text{...}\right)}=\frac{1}{\frac{1}{x^n}+\frac{x}{x^n}+\frac{x^2}{2 x^n}+\frac{x^3}{6 x^n}+\frac{x^4}{24 x^n}+\text{...}}=\frac{1}{\frac{1}{x^n}+\frac{1}{x^{n-1}}+\frac{1}{2 x^{n-2}}+\frac{1}{6 x^{n-3}}+\frac{1}{24 x^{n-4}}+\text{...}}

##

I really don't see how this helps me as I let x go to infinity.

Any suggestions?