# Possible convergence of prime series

## Main Question or Discussion Point

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 pn is the nth prime?

## Answers and Replies

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matt grime
Homework Helper
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
Homework Helper
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.

Anybody else - convergence, divergence or oscillation?

CRGreathouse
Homework Helper
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?

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?

CRGreathouse
Homework Helper
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?
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.

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.

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