- #1

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[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 p

_{n}is the nth prime?

- Thread starter Loren Booda
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- #1

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[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 p

- #2

matt grime

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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.

- #3

CRGreathouse

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[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.

- #4

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Anybody else - convergence, divergence or oscillation?

- #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?Anybody else - convergence, divergence or oscillation?

- #6

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1.

How does the ratio between the product of all even-ordered primes p

2.

How does the ratio between the summation of all even-ordered primes p

- #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.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?

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.

- #8

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- #9

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37 is the number we all find more often then notBut "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.

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