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Math Help - Proving that a_n = (n^3-1)/(n^3+1) converges to 1

  1. #1
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    Proving that a_n = (n^3-1)/(n^3+1) converges to 1

    Hi, I have a very simple example of 1/n converging to 0 (it cannot be solved using a limit, that would be very easy, we have to show it using the definition of the limit), but I don't know how to prove a more complex sequence using the definition of the limit.

    The problem says:

    "Using only the definition of the limit prove that the sequence a_n = (n^3-1)/(n^3+1) converges to 1."

    I would really appreciate it if you could show me how to solve this, thanks a lot!
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  2. #2
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    Re: Proving that a_n = (n^3-1)/(n^3+1) converges to 1

    Given \epsilon, you need to find for which n we have 1 - a_n < \epsilon ( a_n < 1, so this implies |1 - a_n| < \epsilon required by the definition of limit). Just solve this inequality.
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  3. #3
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    Re: Proving that a_n = (n^3-1)/(n^3+1) converges to 1

    note that \frac{n^3-1}{n^3+1} = 1-\frac{2}{n^3+1} so

    |a_n-1| = \frac{2}{n^3+1}

    if we want this to be less than epsilon, how shall we choose n?
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