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Math Help - Limit Proof Question

  1. #1
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    Limit Proof Question

    Hey, will appreciate help with the following Question;
    Prove That,

    \lim \frac{1}{\sqrt{n}} (1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}) = 0

    Thanks in advance for any help !
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  2. #2
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    Alright I think I got it this time:

    0\leq a_n= \frac{1}{\sqrt{n} } \sum_{k=1}^{n} \frac{1}{k} = \frac{1}{n} \sum_{k=1}^{n} \frac{\sqrt{n} }{k} \rightarrow 0 by Cesaro's mean (your sequence is b_n=\frac{\sqrt{n} }{n}=\frac{1}{\sqrt{n} } \rightarrow 0), so a_n \rightarrow 0
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  3. #3
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    I remember that


     \lim_{n\to\infty} \sum_{k=1}^n \frac{a_k}{n} = \lim_{n\to\infty} a_{n}

    this case  a_k = \frac{1}{k}


    so your series

     \lim_{n\to\infty} \frac{1}{\sqrt{n}} \sum_{k=1}^{n} \frac{1}{k}

     =  \lim_{n\to\infty} \frac{\sqrt{n}}{n} \sum_{k=1}^{n} \frac{1}{k}

     =  \lim_{n\to\infty} \sqrt{n} \frac{1}{n}

     = \lim_{n\to\infty} \frac{1}{\sqrt{n}} = 0


    I think this theorem is what Jose27 mentioned ...
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  4. #4
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    did not quite understand

    Hey, thanks for the answer, there is still something unclear to me,
    in order to use cesaro's mean, you assume that
    <br /> <br />
\lim_{n\to\infty}  \sum_{k=1}^{n} \frac{1}{k}<br />
exists ?

    at least by Cauchy Criterion For Convergence (to my knowledge) i find that this sequence does not have a limit.

    i'd appreciate further explanations (also if i got the Cauchy criterion wrong... )

    Thanks and have a nice weekend
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  5. #5
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    Quote Originally Posted by antisane View Post
    Hey, thanks for the answer, there is still something unclear to me,
    in order to use cesaro's mean, you assume that
    <br /> <br />
\lim_{n\to\infty} \sum_{k=1}^{n} \frac{1}{k}<br />
exists ?

    at least by Cauchy Criterion For Convergence (to my knowledge) i find that this sequence does not have a limit.

    i'd appreciate further explanations (also if i got the Cauchy criterion wrong... )

    Thanks and have a nice weekend

    I am not sure what the cesaro's mean is ,

    But talking about the theorem I made use , the series should not have a limit ( diverges to infinty ) , otherwise , the result is to be zero .

     \lim_{n\to\infty} \sum_{k=1}^n \frac{a_k}{n} = \lim_{n\to\infty} \frac{E}{n} = 0

    which is meaningless .
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