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Math Help - sequences/convergence

  1. #1
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    sequences/convergence

    I have one more analysis problem:

    Let t_1 = 1 and t_n+1 = [1 - (1/(n+1)^2)]*t_n for n > 1.

    a) Show that {t_n} converges.
    b) Use induction to show that t_n = (n+1)/2n.
    c) Find the limit of the sequence and prove that it is the limit.
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  2. #2
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    I would start with part b), which you are told to prove by induction. The base case is easy to check, so what about the inductive step? We are told that t_{n+1} = t_n\Bigl(1-\frac1{(n+1)^2}\Bigr), which you can simplify to t_{n+1} = t_n\frac{n(n+2)}{(n+1)^2} (check that!).

    If t_n = \frac{n+1}{2n} then t_{n+1} = \frac{(n+1)n(n+2)}{2n(n+1)^2}, which simplifies to \frac{n+2}{2(n+1)}. That completes the inductive step.

    Once you have done part b), you should be able to work out the limit of t_n as n→∞. That will deal with part a) and also help towards part c).
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