Possible convergence of prime series

[Loren Booda]

Does either

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

or

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

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

 

[matt grime]

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]

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

 

[Loren Booda]

Anybody else - convergence, divergence or oscillation?

 

[CRGreathouse]

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]

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]

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]

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]

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