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Math Help - [SOLVED] Epsilon-N Proof

  1. #1
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    [SOLVED] Epsilon-N Proof

    Prove that the sequence (\frac{1}{1+n+n^4}) converges to 0.

    Here's the solution:



    Here's my problem: I don't understand why they wrote N=\frac{1}{\epsilon} !

    We know that n+1 > \frac{1}{\epsilon}

    Therefore n> \frac{1}{\epsilon} -1

    And so we should take: N= \frac{1}{\epsilon} -1

    Isn't that right???

    Also, I don't understand why they first omitted n^4 from the denominator when they could omit n+1 instead. Because I think \frac{1}{n^4} goes to zero a lot quicker. I appreciate some explanation.
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  2. #2
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    Quote Originally Posted by Roam View Post
    We know that n+1 > \frac{1}{\epsilon}

    Therefore n> \frac{1}{\epsilon} -1

    And so we should take: N= \frac{1}{\epsilon} -1
    No. Firstly, N must be an integer. \frac1\epsilon-1 may not be an integer. Secondly, n+1>\frac1\epsilon is what we want, not what we know. By choosing N=\left\lceil\frac1\epsilon\right\rceil, we can be assured that whenever n>N, we have n+1>\frac1\epsilon.
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  3. #3
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    Quote Originally Posted by proscientia View Post
    \frac1\epsilon-1 may not be an integer.
    Why is that?

    And, what about my second question? Why couldn't we start out by omitting 1+n instead of n^4?

    \frac{1}{1+n+n^4} < \frac{1}{n^4} < \epsilon

    Thus N= \frac{1}{\epsilon ^4}
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  4. #4
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    Quote Originally Posted by Roam View Post
    Thus N= \frac{1}{\epsilon ^4}
    That does not work.
    There is no reason that \frac{1}{\epsilon ^4} is an integer.
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  5. #5
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    Quote Originally Posted by Plato View Post
    That does not work.
    There is no reason that \frac{1}{\epsilon ^4} is an integer.
    Then how do you know that \frac{1}{\epsilon} is an integer?
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  6. #6
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    Where is it stated that \frac1\epsilon is an integer? N is \left\lceil\frac1\epsilon\right\rceil, not \frac1\epsilon.
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  7. #7
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    Quote Originally Posted by Roam View Post
    Then how do you know that \frac{1}{\epsilon} is an integer?
    You don't! If, for example, \epsilon= .00000003= \frac{3}{100000000} then \frac{1}{\epsilon}= \frac{100000000}{3} which is not an integer.

    If you look in any text book you should see that they say something like N> \frac{1}{\epsilon} -1, not N= \frac{1}{\epsilon} -1
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