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Homework Help: Convergence of a sequence

  1. Feb 4, 2010 #1
    Hello, this is a question we had on an exam and I can't figure it out. Our professors won't publish solutions so I'd be glad for your help.
    1. The problem statement, all variables and given/known data
    Prove the following series converges and calculate its limit.

    [tex] 0 < a_0 < \frac {\pi}{2} [/tex]


    [tex]sin(a_n)= \frac {a_n}{a_{n+1}} [/tex]


    and so [tex] 1 > sin(a_0)= \frac {a_0}{a_{1}} [/tex] therefore [tex]a_{1}> a_0 [/tex]

    At first I thought this was simple and the sequence converges to [tex] \frac {\pi}{2} [/tex]
    But I realised that the inequality can hold for any n, i.e [tex] a_{n+1}> a_n [/tex] because we have no way of knowing by how much it is bigger. This one was on our exam and no one I talked to managed to overcome this little detail.
    Some pointers would be greatly apreciated.
    Thanks
    Tal
     
    Last edited: Feb 4, 2010
  2. jcsd
  3. Feb 4, 2010 #2

    Dick

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    Homework Helper

    It does converge to pi/2 and it is pretty simple. Consider the function f(x)=x/sin(x). First figure out what is the range of values of f(x) for x in (0,pi/2). I.e. look for maxs and mins of the function using critical points. Second, show f(x)>x for x in (0,pi/2). If you can show f(x) is bounded by pi/2 for x in (0,pi/2) then a_n is an increasing sequence bounded by pi/2. So a limit exists. If a limit exists then it must satisfy L=L/sin(L). What's the limit?
     
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