Possible convergence of prime series

  • #1
3,077
3

Main Question or Discussion Point

Does either

[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 pn is the nth prime?
 

Answers and Replies

  • #2
matt grime
Science Advisor
Homework Helper
9,395
3
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.
 
  • #3
CRGreathouse
Science Advisor
Homework Helper
2,820
0
My interpretation is
[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
3,077
3
Anybody else - convergence, divergence or oscillation?
 
  • #5
CRGreathouse
Science Advisor
Homework Helper
2,820
0
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?
 
  • #6
3,077
3
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?
 
  • #7
CRGreathouse
Science Advisor
Homework Helper
2,820
0
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.
 
  • #8
3,077
3
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.
 
  • #9
11
0
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
 

Related Threads for: Possible convergence of prime series

Replies
3
Views
3K
Replies
8
Views
4K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
2
Views
2K
Replies
1
Views
2K
  • Last Post
2
Replies
38
Views
5K
  • Last Post
Replies
10
Views
3K
  • Last Post
Replies
9
Views
3K
Top