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Question, I'm new.

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  • Bigfoot
    Is the sum of the reciprocals of all the primes an infinite number? If not what is the approximate number at which it starts trailing off into trivially small
    Message 1 of 4 , Sep 1, 2004
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      Is the sum of the reciprocals of all the primes an infinite number?
      If not what is the approximate number at which it starts trailing off
      into trivially small changes in size? I would assume if the sum of
      all the reciprocals isn't infinite then the product of them wouldn't
      be either?
    • Chris Caldwell
      ... The sum is infinite. It diverges at about the same rate 1/(n*log n) does. See http://www.utm.edu/research/primes/infinity.shtml#converge
      Message 2 of 4 , Sep 1, 2004
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        On Wed, 1 Sep 2004, Bigfoot wrote:
        > Is the sum of the reciprocals of all the primes an infinite number?

        The sum is infinite. It diverges at about the same rate 1/(n*log n) does.
        See http://www.utm.edu/research/primes/infinity.shtml#converge

        > If not what is the approximate number at which it starts trailing off
        > into trivially small changes in size? I would assume if the sum of
        > all the reciprocals isn't infinite then the product of them wouldn't
        > be either?
      • Jud McCranie
        ... Yes, it is. The sum of the reciprocals of the twin primes converges to a finite number, though.
        Message 3 of 4 , Sep 1, 2004
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          At 06:52 PM 9/1/2004, Bigfoot wrote:
          >Is the sum of the reciprocals of all the primes an infinite number?

          Yes, it is. The sum of the reciprocals of the twin primes converges to a
          finite number, though.
        • Nathan Russell
          ... From: Bigfoot Date: Thu, 02 Sep 2004 01:35:31 -0000 Subject: [PrimeNumbers] Re: Question, I m new. To: primenumbers@yahoogroups.com
          Message 4 of 4 , Sep 1, 2004
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            ----- Original Message -----
            From: Bigfoot <plano9@...>
            Date: Thu, 02 Sep 2004 01:35:31 -0000
            Subject: [PrimeNumbers] Re: Question, I'm new.
            To: primenumbers@yahoogroups.com

            What's the method used to determine if a sum or product is infinite
            or not?

            Don't know about products. For sums:

            Generally definite integration works, though it won't tell you
            *exactly* what a value converges to (in which case you use a computer
            to do the first thousand, and can generally see, from what I
            remember).

            In the case of the primes, and some other cases, its outright easy (by
            the way, 1^n+2^n+3^n+... will diverge for all n >= -1, but very
            slowly right at -1)

            Nathan
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