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Math Help - limits of sequences

  1. #1
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    Smile limits of sequences

    Calculate the limits of the sequences:

    A)
    Lim(n→+∞) nCos(n!)/(nē+1)

    B)
    Lim(n→+∞) f(n) , where:
    f(1)=√2
    f(2)=√2√2
    f(3)=√2√2√2 ,....





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  2. #2
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    For your a) question: note that |\cos(n!)|\le 1
    Your other question is un readable.
    Why not learn to post in symbols? You can use LaTeX tags.
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  3. #3
    MHF Contributor chisigma's Avatar
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    Under the hypothesis that the second sequence is...

     a_{0}= 1

    a_{1} = \sqrt{2}

    a_{2} = \sqrt{2 \sqrt{2}}

    \dots

    a_{n+1} = \sqrt {2 a_{n}}

    \dots (1)

    ... the difference equation that defines the sequence is...

    \Delta_{n} = a_{n+1}-a_{n} = \sqrt{2 a_{n}} - a_{n} = f(a_{n}) (2)

    The function f(x)= \sqrt{2x} - x is illustrated here...


    It has a single 'attractive fixed point' in x_{0} = 2 and because \forall x>0 is |f(x)|<|2 - x|, any 'initial value' a_{0}>0 will produce a sequence converging at 2 without oscillations...

    Kind regards

    \chi \sigma
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