1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Power series expansion of a function of x

  1. Feb 19, 2009 #1
    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?


    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.
  2. jcsd
  3. Feb 19, 2009 #2


    User Avatar
    Science Advisor

    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.
  4. Feb 19, 2009 #3


    User Avatar
    Gold Member

    Also the series expansion of [tex]e^{x}[/tex] is valid for all x
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook