Approximating Integral via Power Series

[Amrator]

Homework Statement

Approximate the integral to 3 decimal place accuracy via power series.

$\int_0^{1/2} x^2 e^{-x^2}\, dx$

Homework Equations

The Attempt at a Solution

$x^2 e^{-x^2} = x^2 \sum_{n=0}^\infty \frac {(-x)^{2n}}{n!} = \sum_{n=0}^\infty \frac {x^{2n+2}}{n!}$ ⇒ $\int_0^{1/2} \sum_{n=0}^\infty \frac {x^{2n+2}}{n!}\, dx = \left. \sum_{n=0}^\infty \frac {x^{2n+3}}{(2n+3)n!} \right|_0^{1/2}$

Could I use the ratio test to approximate the error or do I have to use the alternating series test? I don't really see how I could use the alternating series test from here.

 

[vela]

Homework Statement

Approximate the integral to 3 decimal place accuracy via power series.

$\int_0^{1/2} x^2 e^{-x^2}\, dx$

Homework Equations

The Attempt at a Solution

$x^2 e^{-x^2} = x^2 \sum_{n=0}^\infty \frac {(-x)^{2n}}{n!} = \sum_{n=0}^\infty \frac {x^{2n+2}}{n!}$ ⇒ $\int_0^{1/2} \sum_{n=0}^\infty \frac {x^{2n+2}}{n!}\, dx = \left. \sum_{n=0}^\infty \frac {x^{2n+3}}{(2n+3)n!} \right|_0^{1/2}$

Could I use the ratio test to approximate the error or do I have to use the alternating series test? I don't really see how I could use the alternating series test from here.

You messed up the algebra, which is why you don't end up with an alternating series.

 

[Amrator]

I'll go over it again.

 

[Amrator]

I don't see any algebraic mistakes. $(-x)^{2n} = ((-x)^{2})^n = (x^2)^n = x^{2n}$
$x^2x^{2n} = x^{2n+2}$

Edit: Nevermind. I see the algebraic mistake.

 

[geoffrey159]

How do you justify the inversion of $\int$ and $\sum$?

 

[Amrator]

How do you justify the inversion of $\int$ and $\sum$?

What do you mean by inversion?

 

[geoffrey159]

You exchanged $\int$ and $\sum$ in your first post, didn't you ? What is your justification for this ?

 

[Amrator]

You exchanged $\int$ and $\sum$ in your first post, didn't you ? What is your justification for this ?

$e^x = \sum_{n=0}^\infty \frac {x^n}{n!}$
Let $x = -x^2$
Multiply the series by $x^2$, and simply integrate.

 

[geoffrey159]

There is nothing simple about that, you have to prove it

 

[Amrator]

You want me to prove the term by term integration theorem?

 

[geoffrey159]

It's enough if you show that you have all the hypothesis that lead to your conclusion

 

[pasmith]

How do you justify the inversion of $\int$ and $\sum$?

Power series can always be integrated term-by-term within their radius of convergence because convergence of power series is uniform.

(Also the question itself appears to assume that one can do that, so the OP doesn't need to justify it as part of the answer.)

 

[geoffrey159]

I agree, but it must be said somewhere, or proved

 

[Amrator]

I agree, but it must be said somewhere, or proved

I'm in calculus 2, not analysis.

 

[Mark44]

Power series can always be integrated term-by-term within their radius of convergence because convergence of power series is uniform.

(Also the question itself appears to assume that one can do that, so the OP doesn't need to justify it as part of the answer.)

I agree, but it must be said somewhere, or proved

I agree with what pasmith said -- the OP doesn't need to justify interchanging the summation and integration operations.

 

[geoffrey159]

Ok, it's fine for me :-)

@pasmith, convergence of the power serie is uniform on a compact set within the radius of convergence.