Possible convergence of prime series

**Loren Booda** · May24-09, 02:51 AM

Does either

$LaTeX Code: \\frac{\\prod_{2N=n}^\\infty{p_n}}{\\prod_{2N-1=n}^\\infty{p_n}}$

or

$LaTeX Code: \\frac{\\sum_{2N=n}^\\infty{p_n}}{\\sum_{2N-1=n}^\\infty{p_n}}$

converge, diverge or oscillate, where N are the natural numbers, and p_n is the nth prime?

**matt grime** · May24-09, 05:01 AM

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.

**CRGreathouse** · May24-09, 10:28 AM

My interpretation is
$LaTeX Code: \\prod_{n=1}^\\infty\\frac{p_{2n}}{p_{2n-1}}$
which diverges to +infty. But
$LaTeX Code: \\prod_{n=1}^\\infty p_n^{(-1)^n}$
oscillates, so it really depends on how you write it.

**Loren Booda** · May25-09, 08:51 PM

Anybody else - convergence, divergence or oscillation?

**CRGreathouse** · May26-09, 02:04 AM

Originally Posted by Loren Booda

Anybody else - convergence, divergence or oscillation?

Why don't you rewrite it, or explain it in different terms, so we can all be talking about the same thing?

**Loren Booda** · May26-09, 09:33 PM

CRGreathouse,

1.

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

**CRGreathouse** · May27-09, 12:19 AM

Originally Posted by Loren Booda

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

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.

**Loren Booda** · May27-09, 12:47 AM

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.

**John37** · May27-09, 11:08 AM

Originally Posted by CRGreathouse

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.

37 is the number we all find more often then not