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Homework Help: Convergence of a Sequence

  1. Mar 29, 2008 #1
    [SOLVED] Convergence of a Sequence

    1. The problem statement, all variables and given/known data
    Consider the following "recursively defined" sequence:

    a1=0.3
    a(n+1)=sqrt (an+1)

    Compute the first first five terms and prove that it converges. Then, find the limit of the sequence.


    Please see:

    Problem 3, particularly parts c and d here for the complete problem:

    http://rutcor.rutgers.edu/~ngoldberg/math152/ws0837.pdf [Broken]



    3. The attempt at a solution

    I would try to show that the sequence is increasing and bounded because that shows that it converges. I am not sure, however, how to go about doing this.

    I think I am okay with finding the first five terms. I suppose the second term would be a2= sqrt (a1+1)=sqrt (1.3) and so on and so forth and it would get quite complicated.

    I am not sure how to go about the proof and finding the limit.


    Thank you.
     
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Mar 29, 2008 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Welcome to PF!

    Hi workerant! Welcome to PF! :smile:

    Do one step at a time!

    First step is to prove that a_n+1 > a_n.

    In other words, that √(1 + a_n) > a_n

    Hint: what is the condition for √(1 + a) > a? :smile:
     
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