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Math Help - Sequence and limit

  1. #1
    Super Member dhiab's Avatar
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    Thumbs up Sequence and limit

    Calculate limit of every sequence definite in N* by :
    First sequence :



    Second sequence :


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  2. #2
    MHF Contributor Amer's Avatar
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    lim_{n\rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=\left(1+\fra  c{1}{\infty}\right)^{\infty}

    you know that \frac{1}{\infty}=0

    so the final thing will happen

    \left(1+0\right)^{\infty}=1 there is a limit so it is converge

    the second sequence is less than 2

    my solution is wrong so look at whole thread ...........
    Last edited by Amer; June 5th 2009 at 04:43 AM.
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  3. #3
    Super Member dhiab's Avatar
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    Thank you not sure for the first question , more of detail on the second question
    Last edited by dhiab; June 4th 2009 at 10:00 PM.
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  4. #4
    Super Member Random Variable's Avatar
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    The first limit is the definition of e.
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  5. #5
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    Quote Originally Posted by Amer View Post
    \color{red}lim_{n\rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=\left(1+\fra  c{1}{\infty}\right)^{\infty}

    you know that \color{red}\frac{1}{\infty}=0

    so the final thing will happen

    \color{red}\left(1+0\right)^{\infty}=1 there is a limit so it is converge
    That is completely wrong.

    This is a well known limit: \lim _{n \to \infty } \left( {1 + \frac{1}<br />
{n}} \right)^n  = e
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  6. #6
    Super Member Random Variable's Avatar
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    The second one can be proved by induction after you make an educated guess about its limit.
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  7. #7
    MHF Contributor Amer's Avatar
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    Thanks Plato can you explain how it is equal e
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  8. #8
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    That is a common definition of "e".

    You can see a derivation, using a different definition here:http://www.physicsforums.com/showthread.php?t=176076
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