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Homework Statement
[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]
Homework Equations
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.