Convergence of a Prime Sequence

Date: 02/10/2002 at 20:09:14
From: Ken Abbott
Subject: Convergence of a prime sequence?

We know the sum(1/n) (n=1, infinity) does not converge, but what 
about sum(1/p) where sum is over prime numbers only. Does it converge?

Date: 02/10/2002 at 22:44:56
From: Doctor Paul
Subject: Re: Convergence of a prime sequence?

The famous Swiss mathematician Euler found that the series of the 
reciprocals of the primes diverges (although very slowly).

You have to sum 361139 terms of the series just to get the sum to be 
greater than three.

You can read more in Eric Weisstein's World of Mathematics:

   http://mathworld.wolfram.com/PrimeSums.html   

I hope this helps.  Please write back if you'd like to talk about this 
more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/11/2002 at 22:14:06
From: ken abbott
Subject: Convergence of a prime sequence?

Question submitted via WWW:
many thanks for the speedy reply.... I wrote a small Basic program 
and suspected it was diverging. it's very slow... so perhaps it's a 
limiting series of reciprocals, meaning there is no other series of 
reciprocals that diverges slower than this.