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Math Help - power series, w/ interval convergence

  1. #1
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    power series, w/ interval convergence

    express the function as the sum of a power series by first using partial fractions.

    find the interval of convergence.

    heres the function: f(x) = 3/(x^2 + x - 2)


    thankyou for any help with this problem.
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  2. #2
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    Quote Originally Posted by rcmango View Post
    express the function as the sum of a power series by first using partial fractions.

    find the interval of convergence.

    heres the function: f(x) = 3/(x^2 + x - 2)


    thankyou for any help with this problem.
    x^2+x-2 = (x+2)(x-1) thus, f(x) is not defined at -2,1.

    3/(x^2+x-2) = 1/(x-1) - 1/(x+2) for x! = -2 , 1.

    Now,

    1/(x-1) = -1/(1-x) = -1-x-x^2-x^3-... for |x|<1

    1/(x+2) = 1/(1-y) = 1+y+y^2... where y=-1-x and |y|<1

    Thus,

    1/(x+2) = 1-(1+x)+(1+x)^2-(1+x)+... for |1+x|<1

    Our 4 conditions:
    x!=-1 and x!=-2 and |x|<1 and |1+x|<1
    Combine into,
    -1<x<0

    Thus,

    3/(x^2+x-2) = -SUM(n=0,+oo))x^n - SUM(n=0,+oo)(-1)^n(1+x)^n for -1<x<0.
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  3. #3
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    thankyou. clear explanation.

    very nice!
    Last edited by rcmango; April 20th 2007 at 06:21 AM.
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