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Math Help - limit of a sequence

  1. #1
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    limit of a sequence

    [img]http://img708.imageshack.us/img708/4042/analysis999.jpg[/img]
    I stuck halfway through, any help would be appreciated.
    -----------------------------------
    Let ε>0 be given.
    |x_n - (9/2)| = ......
    simplified to:
    39/[(4n^3) + 10]
    Im stuck here, what do i do next?
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by rlkmg View Post
    [img]http://img708.imageshack.us/img708/4042/analysis999.jpg[/img]
    I stuck halfway through, any help would be appreciated.
    -----------------------------------
    Let ε>0 be given.
    |x_n - (9/2)| = ......
    simplified to:
    39/[(4n^3) + 10]
    Im stuck here, what do i do next?
    Well you need to find N in terms of \epsilon

    Note that

    \displaystyle \bigg|\frac{39}{4n^3+10} \bigg|\le \frac{39}{4}\bigg|\frac{1}{n^3} \bigg|

    Now set this equal to epsilon to find big N

    \displastyle \frac{39}{4}\frac{1}{N^3}  =\epsilon \iff N^3=\frac{39}{4\epsilon} \iff N=\sqrt[3]{\frac{39}{4\epsilon}}

    Now you can start the formal proof e.g

    Let \epsilon > 0 and and let  N=\sqrt[3]{\frac{39}{4\epsilon}} then for n > N...
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  3. #3
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    Quote Originally Posted by TheEmptySet View Post
    \displaystyle \bigg|\frac{39}{4n^3+10} \bigg|\le \frac{39}{4}\bigg|\frac{1}{n^3} \bigg|
    Thanks so much, this is the part that I dont understand, could you explain how you got from the left to the right of the above equation? I've seen your technique in many examples but i still dont understand how you did the above step. especially getting rid of the 10

    Thanks for your help, appreciated
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  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by rlkmg View Post
    Thanks so much, this is the part that I dont understand, could you explain how you got from the left to the right of the above equation? I've seen your technique in many examples but i still dont understand how you did the above step. especially getting rid of the 10

    Thanks for your help, appreciated
    okay I think you are over thinking this lets look at a few numerical examples.

    The idea is if you make the denominator of a fraction smaller (divide by a smaller number) the fraction gets bigger for example

    \displaystyle \frac{1}{12}=\frac{1}{2+10} \le \frac{1}{2}

    So the idea is to make the fraction easier to work with by replacing it with a bigger but simpler fraction.
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  5. #5
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    Thanks, i tried a few examples and i've got more of a hang of it
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