How to determine if the series is convergent or divergent.

In summary, the conversation discusses determining if the series \sum x^2e^{-x^2} is convergent or divergent. The attempt at a solution involves using the root test and simplifying the equation, but a mistake was made in the exponent of e which resulted in the incorrect answer. The correct answer is that the series is convergent.
  • #1
Puchinita5
183
0

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?

 
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  • #2


[tex]
(x^2)-(x^2+2x+1)=-(2x+1)
[/tex]

This is the exponent of [tex]e[/tex] on the top, so [tex]e^{2x+1}[/tex] should have been on the bottom.
 
  • #3


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!
!
 

1. How do I determine if a series is convergent or divergent?

To determine if a series is convergent or divergent, you can use various tests such as the Ratio Test, Root Test, and Comparison Test. These tests involve comparing the given series to known convergent or divergent series and analyzing the behavior of the terms in the series.

2. What is the Ratio Test and how does it work?

The Ratio Test is a method for determining the convergence or divergence of a series. It involves taking the limit of the absolute value of the ratio of the (n+1)th term to the nth term as n approaches infinity. If this limit is less than 1, the series is convergent. If it is greater than 1, the series is divergent. If the limit is equal to 1, the test is inconclusive and another test must be used.

3. Can I use the Comparison Test for any series?

The Comparison Test can be used for any series as long as the terms in the series are positive. It involves comparing the given series to a known convergent or divergent series with similar terms. If the known series converges, then the given series also converges. Similarly, if the known series diverges, then the given series also diverges.

4. What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series in which the absolute value of each term is decreasing and the series converges. In contrast, conditional convergence refers to a series in which the absolute value of each term is decreasing but the series alternates between positive and negative terms, causing it to converge to a specific value rather than 0.

5. Why is it important to determine if a series is convergent or divergent?

Determining if a series is convergent or divergent is important because it tells us whether the sum of the terms in the series approaches a finite value or if it diverges to infinity. This information is crucial in applications such as calculating the total energy of a system or estimating the value of a function using a series approximation.

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