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Proving convergence of recursive sequence

  1. Jan 15, 2008 #1
    1. The problem statement, all variables and given/known data

    A sequence is defined recursively by the equations A1 = 1, An+1 = 1/3(An + 4). Show that {An} is increasing and An < 2 for all n. Deduce that {An} is convergent and find its limit.

    2. Relevant equations

    3. The attempt at a solution

    i've put what i've done in this image.
  2. jcsd
  3. Jan 15, 2008 #2
    for n=1 the statement is true
    now suppose it's true for a certain n
    then An+1 = ...<...=2
    here I used the idea that an<2

    Now suppose An+1>An for some n. Use: 1/3(An+4) > An

    Now for n+1, An+2=1/3(An+1 + 4)=1/3(... + 4)=... > 1/3(An+4) if and only if (solve this for An and come to a trivial solution, in example, an<2)

    so now it's increasing and smaller than 2, so...
    For the limit, say an+1=an and solve.
  4. Jan 15, 2008 #3


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    That last statement, "for the limit, say An+1= An and solve" is "shorthand" for what really happens and might be misunderstood (obviously, An+1 is never equal to An). If [itex]\alpha[/itex] is the limit (of course, you must have first shown that the limit exists), taking the limit of both sides of the equation, [itex]A_{n+1}= (1/3)(A_n+ 4)[/itex] to get [itex]lim A_{n+1}= (1/3)(lim A_n+ 4)[/itex] which gives [itex]\alpha= (1/3)(\alpha
  5. Jan 15, 2008 #4
    ofcourse, Halls is right. an+1 is not ever an, but they have the same limit as n becomes really big.
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