- #1

gentsagree

- 96

- 1

## Homework Statement

I ultimately want to discuss convergence of the integral

[tex]\int_{0}^{\infty}\frac{1}{\sqrt{x}e^{\sqrt{x}}}dx[/tex][/B]

## Homework Equations

**[tex]\int_{c}^{\infty}\frac{dx}{x^{p}}[/tex]**

is convergent near x approaching infinity for p>1

3. The Attempt at a Solution

is convergent near x approaching infinity for p>1

3. The Attempt at a Solution

While I understand that the integral converges near 0, as x->infinity I find, for the integrand

[tex]\lim_{x \rightarrow \infty }\frac{1}{\sqrt{x}e^{\sqrt{x}}}=\lim_{x \rightarrow \infty }\frac{1}{\sqrt{x}\left( 1+\sqrt{x} + \frac{x}{2} + \cdots \right) }=\lim_{x \rightarrow \infty }\frac{1}{\left( \sqrt{x} + x + \frac{\sqrt{x}x}{2} + \cdots \right) }[/tex]

but the solution given to me says that I should end up with

[tex]\frac{1}{\left(\frac{\sqrt{x}x}{2} + \cdots \right) }[/tex]

which converges as p would be always greater than 3/2, but I don't see how the limit gets rid of the first two terms in the denominator.

Thanks!