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Evaluating Infinite Series

  1. Mar 11, 2008 #1
    1. The problem statement, all variables and given/known data

    The question is to evaluate the infinite series of the Sum[(((-1)^n)*a(n))/10^n], as n goes from zero to infinity, and a(n) is the recurrence relation a(n)=5a(n-1)-6a(n-2) where a(0)=0, and a(1)=1


    2. Relevant equations

    I found the explicit equation for a(n)=3^n - 2^n, but I can't find how that will help. It doesn't really simplify the sum that I can tell.

    3. The attempt at a solution

    I think that if I could find a generating function for the recurrence relation, then it would probably be a lot easier to relate the series to something that I already know, but I am not sure how to find the generating function. Any help is much appreciated. Thanks a lot.
     
  2. jcsd
  3. Mar 11, 2008 #2

    HallsofIvy

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    Since you know that an= 3n- 2n and obvious thing to do is to put it in the sum- it certainly DOES simplify it!
    The sum becomes
    [tex]\sum_{n=}^{\infty}\frac{(-1)^n(3^n- 2^n)}{10^n}= \sum_{n=0}^\infty\left(\frac{-3}{10}\right)^n}-\sum_{n=0}^\infty\left(\frac{-2}{10}\right)^n[/tex]
    both of which are geometric series.
     
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