How to determine if the series is convergent or divergent.

[Puchinita5]

Homework Statement

Determine if the series is convergent or divergent.
$$\sum x^2e^{-x^2}$$

Homework Equations

The Attempt at a Solution

$$x^2e^{-x^2}=\frac{x^2}{e^{x^2}}$$

$$\lim_{x\to\infty } \frac{(x+1)^2}{e^{(x+1)^2}}\frac{e^{x^2}}{x^2}$$

and since $$(x+1)^2=x^2+2n+1$$

and $$(x^2)-(x^2+2x+1)=-(2x+1)$$

I get $$\lim_{x\to\infty }e^{2x+1}*{(\frac{x+1}{x})}^2=\infty*1=\infty$$ which is $$> 1$$


so by the root test, it is divergent.
Except I got this wrong on my exam. I was told it should be convergent. Why is this wrong?

 

[Sethric]

$$(x^2)-(x^2+2x+1)=-(2x+1)$$

This is the exponent of $$e$$ on the top, so $$e^{2x+1}$$ should have been on the bottom.

 

[Puchinita5]

OMG! i looked at this SO MANY TIMES and didn't see that! thank you! Oh how I love this website!
So it goes to zero, which is less than 1 and so convergent! Glad to know i was doing this right I thought i might have been WAY off!
!