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Math Help - Series Involving Prime Numbers

  1. #1
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    Series Involving Prime Numbers

    Hi, I really don't know how to prove that the following series converges..
    \lim_{x \rightarrow \infty} \sum_{p \leq x}(p \log(p))^{-1}

    Thanks so much!!
    Everk
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  2. #2
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    Re: Series Involving Prime Numbers

    Use the integral test.
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  3. #3
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    Re: Series Involving Prime Numbers

    Quote Originally Posted by ILoveMaths07 View Post
    Use the integral test.
    By the integral test this series diverges.... We have to use much more than this!!!
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  4. #4
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    Re: Series Involving Prime Numbers

    Quote Originally Posted by everk View Post
    Hi, I really don't know how to prove that the following series converges..
    \lim_{x \rightarrow \infty} \sum_{p \leq x}(p \log(p))^{-1}
    I guess you need some heavy machinery for this. The n'th prime number p_n satisfies p_n\geqslant n\log n, because of the prime number theorem. Also (obviously) p_n>n and so \log p_n>\log n.

    Therefore \frac1{p_n\log p_n}<\frac1{n(\log n)^2}. But \sum\frac1{n(\log n)^2} converges, by the integral test. Hence, by the comparison test, so does \sum\frac1{p_n\log p_n}.
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  5. #5
    MHF Contributor chisigma's Avatar
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    Re: Series Involving Prime Numbers

    Quote Originally Posted by everk View Post
    Hi, I really don't know how to prove that the following series converges..
    \lim_{x \rightarrow \infty} \sum_{p \leq x}(p \log(p))^{-1}

    Thanks so much!!
    Everk
    In...

    http://www.mathhelpforum.com/math-he...ers-84832.html

    ... it has been demonstrated that...

    \sum_{k=2}^{n} \pi(k) \sim \ln (\ln n) (1)

    ... where...

    \pi(k) =\begin{cases}\frac{1}{k} &\text{k prime}\\ 0 &\text{elsewhere}\end{cases} (2)

    Now is...

    \ln (\ln n) \sim \int_{2}^{n} \frac{dx}{x\ \ln x} = \ln (\ln n)-\ln (\ln 2) (3)

    ... so that...

    \sum_{k=2}^{n} \frac{\pi(k)}{\ln k} \sim \int_{2}^{n} \frac{dx}{x\ \ln^{2} x} = \frac{1}{\ln 2}-\frac{1}{\ln n} (4)

    Kind regards

    \chi \sigma
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