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Homework Help: Converging sequence

  1. Feb 27, 2007 #1
    1. The problem statement, all variables and given/known data

    Let t1=sqrt(2)

    Let t_(n+1)=sqrt(2+sqrt(t_n)) (it's a recursively defined series)

    What does it converge to?


    2. Relevant equations



    3. The attempt at a solution

    I calculated it out for some values andI get 1.8312 (approx), but I don't want to express it in decimals, and I want to know if there's a good way to do this.
     
  2. jcsd
  3. Feb 27, 2007 #2
    Let's suppose it converges, then t_(n+1) and t_n will have a infinitesimally small difference as n approaches infinity, and you can consider them the same number. (aka the limit of this sequence)
     
  4. Feb 28, 2007 #3

    HallsofIvy

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    Science Advisor

    I would have phrased it differently, but basically said the same thing. Since {tn+1[/b]} is exactly the same sequence as {tn}, just indexed differently, taking the limit as n goes to infinity gives the same value, say T, on both sides. Solve that equation for T. (That's not a trivial equation but there is one obvious root!)
     
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