# Homework Help: Comparison theorem

1. Dec 9, 2007

### fk378

1. The problem statement, all variables and given/known data
Use the Comparison Theorem to determine whether the integral below is convergent or divergent:

e^-x / sqrt x dx integrated from 0-->1

3. The attempt at a solution
I think it is convergent because 1/e^x is convergent. I set the original integral less than or equal to the integral of 1/(e^x) dx

When I solved for it, I got -1/e + 1, therefore it is convergent. However, my professor marked my paper as saying it's not true. He set the original integral less than or equal to 1/sqrt x, and solving for that, got 2. Why can't my comparison hold true?

2. Dec 9, 2007

### quasar987

Because it's not true in (0,1] that e^-x / sqrt x <e^-x

3. Dec 9, 2007

### Kreizhn

Your comparison doesn't hold true because on (0,1), we have that

$$\frac{e^{-x}}{\sqrt{x}} > e^{-x}$$

This follows from the fact that on (0,1), $\sqrt{x}< 1 \Rightarrow \frac{1}{\sqrt{x}} > 1}$

4. Dec 9, 2007

### Kreizhn

Haha, barely beaten to it. I knew I shouldn't have wasted my time previewing the post :tongue: