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Math Help - Cauchy Sequences

  1. #1
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    Cauchy Sequences

    1. Can anyone provide an example of a sequence that satisfies the following definition, but it not Cauchy:

      for every e>0 there exists natural number N such that for every n>or=N |xn-xn+1|<e
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by amm345 View Post
    1. Can anyone provide an example of a sequence that satisfies the following definition, but it not Cauchy:

      for every e>0 there exists natural number N such that for every n>or=N |xn-xn+1|<e
    How about the sequence of partial sums H_n=\sum_{k=1}^{n}\frac{1}{k}. It's not Cauchy since it's divergent, but H_{n+1}-H_n=\frac{1}{n+1}, which can be made as small as desired.
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