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Possible convergence of prime series

  1. May 24, 2009 #1
    Does either

    [tex]\frac{\prod_{2N=n}^\infty{p_n}}{\prod_{2N-1=n}^\infty{p_n}}[/tex]

    or

    [tex]\frac{\sum_{2N=n}^\infty{p_n}}{\sum_{2N-1=n}^\infty{p_n}}[/tex]

    converge, diverge or oscillate, where N are the natural numbers, and pn is the nth prime?
     
  2. jcsd
  3. May 24, 2009 #2

    matt grime

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    Assuming we do all the cancellation possible in the first one without worrying what it means, and that 2N=n really ought to be written n=2N, then it simplifies to

    1/p_{2N-1}

    which converges to 0 as N tends to infinity.

    I don't think N can mean the natural numbers by the way.
     
  4. May 24, 2009 #3

    CRGreathouse

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    My interpretation is
    [tex]\prod_{n=1}^\infty\frac{p_{2n}}{p_{2n-1}}[/tex]
    which diverges to +infty. But
    [tex]\prod_{n=1}^\infty p_n^{(-1)^n}[/tex]
    oscillates, so it really depends on how you write it.
     
  5. May 25, 2009 #4
    Anybody else - convergence, divergence or oscillation?
     
  6. May 26, 2009 #5

    CRGreathouse

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    Why don't you rewrite it, or explain it in different terms, so we can all be talking about the same thing?
     
  7. May 26, 2009 #6
    CRGreathouse,

    1.

    How does the ratio between the product of all even-ordered primes pn (n=2N; n=2, 4, 6...) and the product of all odd-ordered primes pn (n=2N-1; n=1, 3, 5...) behave as n approaches infinity?

    2.

    How does the ratio between the summation of all even-ordered primes pn (n=2N; n=2, 4, 6...) and the summation of all odd-ordered primes pn (n=2N-1; n=1, 3, 5...) behave as n approaches infinity?
     
  8. May 26, 2009 #7

    CRGreathouse

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    But "the product of all even-ordered primes" is infinite, as is "the product of all odd-ordered primes". You can't sensibly take the ratio at all.

    I gave two ways (post #5) to do the operation: take factors two at a time:
    (3/2) * (7/5) * (13/11) * ...
    which diverges, and taking them one factor at a time:
    (1/2) * 3 * (1/5) * 7 * (1/11) * ...
    which may oscillate.

    But you may intend neither of these; that's why I asked for clarification.
     
  9. May 26, 2009 #8
    You reminded me of the book Gamma by Julian Havil [p. 22-24] that the apparent behavior of an infinite calculation may contradict itself according to how its terms are grouped - like you say, as is written.
     
  10. May 27, 2009 #9
    37 is the number we all find more often then not
     
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