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Math Help - limit and summation help

  1. #1
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    limit and summation help

    Prove that the limit as p approaches infinity of the summation from k=1 to infinity of 1/k^p =1 and the limit as p approaches 1+ of the summation from k=1 to infinity of 1/k^p= infinity.

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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by friday616 View Post
    Prove that the limit as p approaches infinity of the summation from k=1 to infinity of 1/k^p =1 and the limit as p approaches 1+ of the summation from k=1 to infinity of 1/k^p= infinity.

    Thanks!
    I helped you with two other ones. Now, by George, I want to see some effort!
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  3. #3
    MHF Contributor chisigma's Avatar
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    Lets call \zeta(p) the function...

    \zeta (p) = \lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{1}{i^{p}} (1)

    First we prove that the limit (1) is \infty if p=1. At this scope we compute the partial sum for n=2^{k} of (1) for p=1...

    \sum_{i=1}^{2^{k}} \frac{1}{i}= 1 + \frac{1}{2} + \{\frac{1}{3} + \frac{1}{4}\} + \{\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}\} + \dots +\{\frac{1}{2^{k-1}+1} + \frac{1}{2^{k-1}+2} +\dots + \frac{1}{2^{k}-1} +\frac{1}{2^{k}} \} (2)

    ... and it is easy to see that is...

     \sum_{i=1}^{2^{k}} \frac{1}{i} > 1 + \frac {k}{2} \rightarrow \lim_{k \rightarrow \infty} \sum_{i=1}^{2^{k}} \frac{1}{i} = \infty (3)

    Lets now compute the partial sum for n=2^{k} of (1) for p>1...

    \sum_{i=1}^{2^{k}} \frac{1}{i^{p}}= 1 + \frac{1}{2^{p}} + \{\frac{1}{3^{p}} + \frac{1}{4^{p}}\} + \{\frac{1}{5^{p}} + \frac{1}{6^{p}} + \frac{1}{7^{p}} + \frac{1}{8^{p}}\} + \dots

    \dots +\{\frac{1}{(2^{k-1)}+1)^{p}} + \frac{1}{(2^{k-1}+2)^{p}} +\dots + \frac{1}{(2^{k}-1)^{p}} +\frac{1}{2^{kp}} \} (4)

    ... and it is easy to see that is...

     \sum_{i=1}^{2^{k}} \frac{1}{i^{p}} < 1 + (\frac{1}{2})^{p} + (\frac{1}{2})^{2p} + \dots + (\frac{1}{2})^{(k-1)p} \rightarrow \sum_{i=1}^{\infty} \frac{1}{i^{p}} < \infty (5)

    The results obatined in (3), (4) and (5) allow us to conclude that is...

    \lim_{p \rightarrow 1+} \sum_{i=1}^{\infty} \frac{1}{i^{p}}= \infty

    \lim_{p \rightarrow \infty} \sum_{i=1}^{\infty} \frac{1}{i^{p}}=1 (6)

    Kind regards

    \chi \sigma
    Last edited by chisigma; November 21st 2009 at 06:03 AM.
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