# Power series expansion of a function of x

#### Unassuming

1. The problem statement, all variables and given/known data

[Directions to problem]
Show that the function of x gives a power series expansion on some interval centered at the origin. Find the expansion and give its interval of validity.

$$\int_0^x e^{-t^2} dt$$

2. Relevant equations

3. The attempt at a solution

I have that, $$e^{-t^2} = \sum_0^{\infty} \frac{(-1)^n(t^2)^n}{n!}$$

I now am wondering whether I can take the integral of this series as follows,

$$\int_0^x \sum_0^{\infty} \frac{(-1)^n(t^2)^n}{n!} dt = \sum_0^{\infty} \frac{(-1)^n(x^{2n+1})}{n!(2n+1)}$$

Am I allowed to do that and if so, what is the justication?

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I performed the ratio test on the result and the limit as n approached 0 was 0, and I therefore concluded that the series converges for all x in R.

#### HallsofIvy

I think you mean "prove the function has a power series" since the integration itself does not directly "give" the power series.

Yes, as long as you are inside the radius of convergence, you can integrate a power series term by term.

#### djeitnstine

Gold Member
Also the series expansion of $$e^{x}$$ is valid for all x

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