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Math Help - Showing the sequence is bounded.

  1. #1
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    Showing the sequence is bounded.

    Hey all i thought i understood this question, but after having another look i think my solution is wrong.

    "Show that the sequence {an} is bounded, where an, is given by

     <br />
a_n = \frac {(-1)^n}{n^3}<br />
 "

    I know what if i show {an} is convergent, then it is therefore bounded.

    My initial idea was to show that an was convergent by the alternating series test.

    But thats the problem, its not a series.
    So is my method correct, or shall i be approaching this differently?

    Thanks for your help!
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  2. #2
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    What am I missing?
    \left| {\frac{{\left( { - 1} \right)^n }}<br />
{{n^3 }}} \right| = \frac{1}<br />
{{n^3 }} \leqslant 1<br />
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  3. #3
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    Quote Originally Posted by Plato View Post
    What am I missing?
    \left| {\frac{{\left( { - 1} \right)^n }}<br />
{{n^3 }}} \right| = \frac{1}<br />
{{n^3 }} \leqslant 1<br />
    Why did u just take the absolute value of an?
    And i dont get what your asking me?
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  4. #4
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    Quote Originally Posted by simpleas123 View Post
    Why did u just take the absolute value of an?
    And i dont get what your asking me?
    I don't get what it is that you do not understand.
    Do you understand what it means to be bounded?
    Last edited by Plato; May 14th 2010 at 10:08 AM.
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  5. #5
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    Quote Originally Posted by Plato View Post
    I don't get what it is that you do not understand.
    Do you understand what it means to be bounded?
    Ahh, its so simple!
    <br />
| a_n | \leqslant M<br />
    For some M.
    In this case, being that an converges to 1 therefore M = 1.

    Thank you very much!
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  6. #6
    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by simpleas123 View Post
    Ahh, its so simple!
    <br />
| a_n | \leqslant M<br />
    For some M.
    In this case, being that an converges to 1 therefore M = 1.

    Thank you very much!
    a_n does not converge to 1.
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  7. #7
    Member mabruka's Avatar
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    Leave alone for a moment the statement: "If (a_n)_n is convergent then it is bounded".


    Boundedness is a concept that comes before convergence,


    "A sequence (a_n)_n is bounded if there is a number M such that

    |a_n|\leq M for all n.


    Convergence has nothing to do with boundedness so far.


    Thats why Plato did what he did.
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