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Math Help - sequence question

  1. #1
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    sequence question

    Could someone show me what to do here:
    For each positive integer n, let fn be the function defined by
    fn(x) = x^2n + x^(2n−1) + ... + x^2 + x + 1. Let mn be the minimum of fn on the interval [−1,0]. Prove that the sequence (mn) converges to a limit A and then find A.

    Thank you for your help.
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  2. #2
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    Quote Originally Posted by PvtBillPilgrim View Post
    Could someone show me what to do here:
    For each positive integer n, let fn be the function defined by
    fn(x) = x^2n + x^(2n−1) + ... + x^2 + x + 1. Let mn be the minimum of fn on the interval [−1,0]. Prove that the sequence (mn) converges to a limit A and then find A.

    Thank you for your help.
    We can wrtite x^{2n}+x^{2n-1}+...+x+1 = \frac{1-x^{2n+1}}{1-x} \mbox{ if }x\not = 1 \mbox{ and }2n+1 \mbox{ if } x=1. So the mimmum occurs for zero for all n.

    I think you missted something because it is not a very interesting problem.
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  3. #3
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    Is what you did correct?

    f_n+1(x) - f_n(x) is decreasing (I believe). This should imply that (m_n) is decreasing. And it is bounded by [-1,0] so it must be convergent.

    As n goes to infinity, f_n(x) goes to 1/(1-x) on (-1,0]. How do I find the limit A though (to what (m_n) converges)?

    Thanks for your help.
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  4. #4
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    Quote Originally Posted by PvtBillPilgrim View Post
    Is what you did correct?

    f_n+1(x) - f_n(x) is decreasing (I believe). This should imply that (m_n) is decreasing. And it is bounded by [-1,0] so it must be convergent.

    As n goes to infinity, f_n(x) goes to 1/(1-x) on (-1,0]. How do I find the limit A though (to what (m_n) converges)?

    Thanks for your help.
    I make a minor mistake. I did it on the interval [0,1] you wanted on the interval [-1,0]. But can you use the same approach and do the problem yourself?
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  5. #5
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    Well do you know what the limit A is?

    I think it may be 1/2 just looking at my limit above as n goes to infinity. Does this sound right?
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  6. #6
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    I did not do the limit. I told you these polynomial can be written as f(x) = \frac{1-x^{2n+1}}{1-x}. Now how do you find the minimum? Find the derivative and make it zero. Can you do that? That will be your sequence.
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