Prime Divergence

December 12, 2012

Did you know that:

\sum_{n=prime}^{ \infty}=\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{prime_n}+...

is equal infinity? Now, some of you might be thinking, “Oh, I already knew that.” Well, then this post is not for you, move along. Others, though, might think, “hold on, isn’t adding anything up to infinity equals infinity?” Nope because the sum (1/2)+(1/4)+(1/8)+ etc for example is equal to 1. And in fact, all geometric sum (in case you don’t know, it goes like n+n^2+n^3+… etc) with a number less than one converge into a number that is not infinite.

Some others who has learned stuff from second semester calculus class might think “Really? It looks real slow.” Well, it is really slow, but as you shall see, it nevertheless does diverge, and demonstrating this will be the topic of this post.

This post is based on a section in the book “An Imaginary Tale” by Paul Nahin. The book talks about this, and I thought the mathematics was super neat, and so I decided to put it here.

The post is in three parts:

a)I will show a proof originally done by Nicole Oresme (1320-1382) on the divergence of the harmonic series, which is 1+(1/2)+(1/3)+…

b)Then I will change the Riemann zeta function, which is f(x)=1+(1/2^x)+(1/3^x)… into a product of primes, which was done by Leonard Euler (1707-1783).

c)And finally I will use the above results to prove that the sum of all 1/prime does diverge, which was proved by Euler. Read the rest of this entry »