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Finding the radius of convergence of a power series

  1. Jul 18, 2015 #1
    1. The problem statement, all variables and given/known data
    Σ(n=0 to ∞) ((20)(-1)^n(x^(3n))/8^(n+1)

    2. Relevant equations
    Ratio test for Power Series: ρ=lim(n->∞) a_(n+1)/a_n

    3. The attempt at a solution
    I tried the ratio test for Power Series and it went like this:

    ρ=lim(n->∞) (|x|^(3n+1)*8^(n+1))/(|x|^(3n)*8^(n+2))
    =20|x|/8 lim(n->∞) 1
    =20|x|/8

    20|x|/8<1
    |x|<2/5

    So radius of convergence=2/5.


    However when I input the power series into Mathematica, it says that the radius of convergence should be 2 (http://goo.gl/9nAHoS)

    Where am I going wrong in my calculation?
     
  2. jcsd
  3. Jul 18, 2015 #2

    RJLiberator

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    Gold Member

    Hi there mate, I am going to attempt to help.

    Here are some possible errors that I am looking into:

    First, x^(3n) when you make it n+1 in the numerator of the ratio test, it should be x^(3n+3), correct?

    Well, that might do it. Does this help?
     
  4. Jul 18, 2015 #3

    Ray Vickson

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    Try setting ##x^3 = t## and finding the radius of ##t##-convergence for ##\sum_n c_n t^n##. Then translate those results into statements about ##x##. Alternatively, do it over again, but repair the algebraic errors you made. (Avoidance of such errors is the reason I suggested looking at ##t## instead of ##x##.)
     
  5. Jul 18, 2015 #4

    RJLiberator

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    Also, one other thing --> In your solution you have a 20 sticking out, however, it seems to me that when you take the a^(n+1) term and the a^n term and divide them, the 20 on each will cancel leaving |x^3|/8 <1
     
  6. Jul 18, 2015 #5
    @RJLiberator and @Ray Vickson, thank you both for the help. Sloppy algebra seems to be my perpetual downfall.
     
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