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Power Series

  1. Mar 22, 2009 #1
    1. The problem statement, all variables and given/known data
    Determine the series of the given function:

    f(x) = 10 / (1-5*x)



    2. Relevant equations

    Power series of 1/(1-x) = Σ from n=0 to n=infinity of (x^n)


    3. The attempt at a solution

    f(x) = 10/(1-5x)
    = 10*(1/1-5x)
    = 10 * Σ(5x)^n
    = 10 * Σ(5^n)*(x^n)
    = Σ (50^n)*(x^n) <--- Not sure if that is right

    Any help would be appreciated. Thank you
     
  2. jcsd
  3. Mar 22, 2009 #2

    Dick

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    Not right. The series part is fine. But 10*(5^n) does not equal 50^n. Think about, say, n=2.
     
  4. Mar 22, 2009 #3
    Is it just Σ(5^n)*(x^n)*10 ?
     
  5. Mar 22, 2009 #4

    Dick

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    I think that's the simplest way to write it, yes.
     
  6. Mar 22, 2009 #5
    One more thing. To find the interval on convergence, I know I have to take the ratio test as n-->infinity. Is this how I'm supposed to set it up?

    lim [x^(n+1) * 5^(n+1) / (n+1)*(n+1)] * [(n*n/x^n*5^x)]
    n->inf

    After canceling out some factors I got:

    lim 1/(2n+1) = 0
    n->inf

    Is that right?
     
  7. Mar 22, 2009 #6

    Dick

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    No. Where are all those n+1 and n's coming from? The nth term of your series a_n=10*5^n*x^n. So the ratio of a_(n+1)/a_n is just 10*5^(n+1)*x^(n+1)/(10*5^n*x^n) isn't it? What's that?
     
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