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Math Help - limit

  1. #1
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    limit

    (x_n)_{n\geq 1}, x_1 \in R, x_{n + 1} = \frac {2}{1 + x_n^2}, n \geq 1. Prove that \lim_{n\to \infty} = 1
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  2. #2
    Super Member Deadstar's Avatar
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    You want to solve...

    L = \frac{2}{1 + L^2} where L stands for the limit.

    This gives L^3 + L - 2 = 0, solve this to find your limit. (factor by (x-1))
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  3. #3
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    But I don't need to prove that it is convergent(it has a limit)?
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  4. #4
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    Quote Originally Posted by ely_en View Post
    But I don't need to prove that it is convergent(it has a limit)?
    Yes, you do need to prove that the limit exists. You probably can do that by proving (by induction) that the sequence is increasing and has an upper bound (if x_1 is less than that limit) or is decreasing and has a lowwer bound (if x_1 is greater than that limit).
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