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Find the limit of a recursive sequence

  1. Apr 22, 2012 #1
    1. The problem statement, all variables and given/known data
    Suppose |r|<1

    Find the limit of {|r|^n} by treating it as a recursive sequence defined by x_1=|r| and x_n=|r|*x_(n-1)


    2. Relevant equations
    This is proving a theorem in the book:

    If |r| < 1, then the sequence {r^n} converges to 0.


    3. The attempt at a solution

    It is clear that without the theorem this sequence converges to 0. Showing it is the issue, so maybe it is not so clear?

    This is part 2, in part 1 I showed it converges by the monotone convergence theorem, so the limit does exist.

    Suppose x_n -> L for some L in ℝ.

    Since [itex]lim_{n->∞}[/itex]x_n = L, then [itex]lim_{n->∞}[/itex]x_n+[itex]lim_{n->∞}[/itex]x_(n+1) = 2L

    Which this is the same as saying [itex]lim_{n->∞}[/itex]x_n+|r| [itex]lim_{n->∞}[/itex]x_n = 2L

    => L+|r|L = 2L => L+|r|L-2L = 0 => -L+|r|L = 0 => L(-1+|r|)= 0 => L = 0

    So the sequence converges to 0.


    Am I correct or did I do something wrong?
     
  2. jcsd
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