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Math Help - cauchy sequence

  1. #1
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    cauchy sequence

    Show directly that the following sequence is not cauchy :
    x_n=ln \ n
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    Senior Member Shanks's Avatar
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    Hint:what is \left|x_{2n}-x_{n}\right|?
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    Quote Originally Posted by flower3 View Post
    Show directly that the following sequence is not cauchy :
    x_n=ln \ n
    It is easy to show that if N\in\mathbb{Z}^+~\&~k\in\mathbb{Z}^+ then \frac{k}{N+k}\le \ln(N+k)-\ln(N).
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by flower3 View Post
    Show directly that the following sequence is not cauchy :
    x_n=ln \ n
    Quote Originally Posted by Shanks View Post
    Hint:what is \left|x_{2n}-x_{n}\right|?
    I think what Shanks meant was consider that for every n>1 it is obviously true that n^2>n, but \left|\ln\left(n^2\right)-\ln(n)\right|=\left|2\ln(n)-\ln(n)\right|=\left|\ln(n)\right|=\ln(n), which is clearly greater than any \varepsilon>0 for sufficiently large n.
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  5. #5
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    Quote Originally Posted by flower3 View Post
    Show directly that the following sequence is not cauchy :
    x_n=ln \ n
    cancel this post
    Last edited by xalk; December 20th 2009 at 09:48 AM.
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