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Thread: help with limits of sequences

  1. #1
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    help with limits of sequences

    Hey everyone

    I am a bit confuse about the following questions especially about the first and the second one

    help with limits of sequences-question2.png

    I need to find each sequence's limit and it's pretty urgent

    please I really need help with those three

    thanks
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  2. #2
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    Re: help with limits of sequences

    $$\left(1-\frac1{n^2}\right)^n = \left(1-\frac1{n^2}\right)^{(n^2\frac{n}{n^2})} = \left(\left(1-\frac1{n^2}\right)^{n^2}\right)^{\frac1n}$$
    The expression inside the (outer) brackets is easily transformed into a standard result.

    Similar manipulations deal with the other examples. For the second, write $$n^2+1=(n^2-1)+2$$
    Last edited by Archie; Dec 30th 2017 at 06:16 PM.
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  3. #3
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    Re: help with limits of sequences

    Quote Originally Posted by someone111888 View Post
    the first and the second one
    Click image for larger version. 

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    I need to find each sequence's limit and it's pretty urgen
    I suggest for the second one you might consider:
    ${\left( {\dfrac{{{n^2} + 1}}{{{n^2} - 1}}} \right)^{{n^2} + 3n - 16}} = {\left( {1 + \dfrac{2}{{{n^2} - 1}}} \right)^{{n^2}}}{\left( {1 + \dfrac{2}{{{n^2} - 1}}} \right)^{3n - 16}}$
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  4. #4
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    Re: help with limits of sequences

    but how can I solve it from here?
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  5. #5
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    Re: help with limits of sequences

    Quote Originally Posted by someone111888 View Post
    but how can I solve it from here?
    Suppose $\displaystyle S = \lim_{n\to \infty}s_n $.

    Then $\displaystyle \ln S = \ln \lim_{n\to \infty} s_n = \lim_{n\to \infty} \ln (s_n) $
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