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Homework Help: Limit of sequence

  1. Aug 27, 2008 #1
    1. The problem statement, all variables and given/known data
    an+1=an/2 + 1/an

    Prove that the above sequence converge and find the limit.
    2. Relevant equations



    3. The attempt at a solution
    I have used Maple 12 to compute up to 10 term, using different initial value of a0. I found that the sequence is approaching square root of two when a0 is positive, and negative of square root of positive when a0 is negative. From the software, I know that the sequence is convergent, but it is rather difficult to find a mathematical proof.

    Is it reasonable to say that an approximate to an+1 when n is approaching infinity? If so how can I prove it? Can I substitute both an and an+1 with ainfinity and find the limit?

    Thanks in advanced.
     
  2. jcsd
  3. Aug 27, 2008 #2
    If you know that your thus defined sequence [itex](a_n)_{n\geq0}[/itex] converges to some number c then the sequence [itex](a_{n+1})_{n\geq0}[/itex] converges to that same number.
    Then you can use the continuity of the function [itex]x\mapsto \frac{x}{2}+\frac{1}{x}[/itex] to take the limit of the equation [itex]a_n=\frac{a_n}{2}+\frac{1}{a_n}[/itex] to obtain [itex]c=\frac{c}{2}+\frac{1}{c}[/itex] which has the solution [itex]c=\sqrt{2}[/itex] of course.

    So if the sequence is convergent the limit is [itex]\sqrt{2}[/itex].

    But in order to conclude that your sequence converges in the first place you need to work some more. How would you prove in general that a sequence is convergent?
     
  4. Aug 27, 2008 #3

    HallsofIvy

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    Well, you shouldn't call it ainfinity. Rather, IF the sequence converges, say to some number A, then, taking limits on both sides of the equation A= A/2+ 1/A. From that, multiplying on both sides by A, A2+ A2/2+ 1 or (1/2)A2= 1 or A2= 2. Assuming that a0> 0 then [itex]A= \sqrt{2}[/itex].

    That is, of course, IF the sequence converges. With that information, we can then use "monotone convergence" to show that it does, in fact, converge.
     
    Last edited by a moderator: Aug 27, 2008
  5. Aug 27, 2008 #4

    statdad

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    Homework Helper

    As a slight (I hope not too far-reaching) expansion on the previous correct ideas:

    1. Try to show that the sequence is non-decreasing (i.e. [tex] a_{n+1} \ge a_n[/tex])
    2. Show that the sequence is bounded above ([tex] a_n \le M [/tex] for some
    positive number [tex] M [/tex])

    These two steps together will be enough to prove that the sequence converges; you've already seen how to find the limit.
     
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