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**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.

[tex] \int_0^x e^{-t^2} dt [/tex]

**2. Relevant equations**

**3. The attempt at a solution**

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

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

[tex]\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)} [/tex]

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.