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Limit of a sequence

  1. Jul 26, 2008 #1
    1. The problem statement, all variables and given/known data
    Find the limit of the sequence
    { sqrt(2), sqrt(2sqrt(2)), sqrt(2sqrt(2sqrt(2))) ... }

    2. Relevant equations
    Limit Laws?


    3. The attempt at a solution
    I wrote out the first five values in the sequence and came to the conclusion that this sequence could be written out as

    [tex]A_{n} = 2^\frac{2^{n}-1}{2^{n}}[/tex]

    I then took [tex]\frac{2^{n}-1}{2^{n}}[/tex], broke it down to [tex]1 - \frac{1}{2^{n}}[/tex] which allowed me to rewrite the equation to [tex]2\times2^\frac{-1}{2^{n}}[/tex]. Ignoring the 2 for now, I re-worked the fraction exponent and resulted with [tex]-(\frac{1}{2})^{n}[/tex] and made the value into a fraction [tex]\frac{1}{2^(\frac{1}{2})^{n}}[/tex].

    Using the sheer power of what is known as the graphing calculator, I was able to determine that the limit of that equation is 1, and then multiplying 2 to it gave 2. Without a calculator, how can I lay out the steps?

    Note to self: BUY A TABLET!!
     
    Last edited: Jul 26, 2008
  2. jcsd
  3. Jul 26, 2008 #2

    Borek

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    Staff: Mentor

    Won't it be enough to calculate limit of

    [tex]1 - \frac 1 {2^n}[/tex]

    Seems rather obvious. But then I am mathematically challenged and could be I am missing some fine print.
     
  4. Jul 26, 2008 #3

    Defennder

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    Homework Helper

    Yeah I believe Borek is right.
     
  5. Jul 26, 2008 #4
    I will outline a solution. The details are your job.

    There's a much simpler way of writing the sequence as a recurrence relation. Use this way.

    First show that each term is less than a certain constant. Next demonstrate that the sequence is increasing. Hence show the sequence is convergent with some undetermined limit L.

    The continuity of a certain function (which one?) will allow you to take limits of both sides of the recurrence relation.

    The exact value of the limit L should now be in sight. I'll leave the rest up to you.

    I will outline a solution. The details are your job.

    There's a much simpler way of writing the sequence as a recurrence relation. Use this way.

    First show that each term is less than a certain constant. Next demonstrate that the sequence is increasing. Hence show the sequence is convergent with some undetermined limit L.

    The continuity of a certain function (which one?) will allow you to take limits of both sides of the recurrence relation.

    The exact value of the limit L should now be in sight. I'll leave the rest up to you.

    If you're up for the challenge, you might also try to find the set of all x such that the sequence {x, x^x, x^(x^x), ...} is convergent.
     
    Last edited: Jul 26, 2008
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