Question, I'm new.

Expand Messages
• Is the sum of the reciprocals of all the primes an infinite number? If not what is the approximate number at which it starts trailing off into trivially small
Message 1 of 4 , Sep 1, 2004
• 0 Attachment
Is the sum of the reciprocals of all the primes an infinite number?
If not what is the approximate number at which it starts trailing off
into trivially small changes in size? I would assume if the sum of
all the reciprocals isn't infinite then the product of them wouldn't
be either?
• ... The sum is infinite. It diverges at about the same rate 1/(n*log n) does. See http://www.utm.edu/research/primes/infinity.shtml#converge
Message 2 of 4 , Sep 1, 2004
• 0 Attachment
On Wed, 1 Sep 2004, Bigfoot wrote:
> Is the sum of the reciprocals of all the primes an infinite number?

The sum is infinite. It diverges at about the same rate 1/(n*log n) does.
See http://www.utm.edu/research/primes/infinity.shtml#converge

> If not what is the approximate number at which it starts trailing off
> into trivially small changes in size? I would assume if the sum of
> all the reciprocals isn't infinite then the product of them wouldn't
> be either?
• ... Yes, it is. The sum of the reciprocals of the twin primes converges to a finite number, though.
Message 3 of 4 , Sep 1, 2004
• 0 Attachment
At 06:52 PM 9/1/2004, Bigfoot wrote:
>Is the sum of the reciprocals of all the primes an infinite number?

Yes, it is. The sum of the reciprocals of the twin primes converges to a
finite number, though.
• ... From: Bigfoot Date: Thu, 02 Sep 2004 01:35:31 -0000 Subject: [PrimeNumbers] Re: Question, I m new. To: primenumbers@yahoogroups.com
Message 4 of 4 , Sep 1, 2004
• 0 Attachment
----- Original Message -----
From: Bigfoot <plano9@...>
Date: Thu, 02 Sep 2004 01:35:31 -0000
Subject: [PrimeNumbers] Re: Question, I'm new.

What's the method used to determine if a sum or product is infinite
or not?

Don't know about products. For sums:

Generally definite integration works, though it won't tell you
*exactly* what a value converges to (in which case you use a computer
to do the first thousand, and can generally see, from what I
remember).

In the case of the primes, and some other cases, its outright easy (by
the way, 1^n+2^n+3^n+... will diverge for all n >= -1, but very
slowly right at -1)

Nathan
Your message has been successfully submitted and would be delivered to recipients shortly.