- #1
Puchinita5
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Homework Statement
Determine if the series is convergent or divergent.
[tex]\sum x^2e^{-x^2}[/tex]
Homework Equations
The Attempt at a Solution
[tex]
x^2e^{-x^2}=\frac{x^2}{e^{x^2}}[/tex]
[tex]\lim_{x\to\infty } \frac{(x+1)^2}{e^{(x+1)^2}}\frac{e^{x^2}}{x^2}[/tex]
and since [tex] (x+1)^2=x^2+2n+1 [/tex]
and [tex](x^2)-(x^2+2x+1)=-(2x+1)[/tex]
I get [tex]\lim_{x\to\infty }e^{2x+1}*{(\frac{x+1}{x})}^2=\infty*1=\infty[/tex] which is [tex] > 1[/tex]
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?