We know that the harmonic series 1
s. On the other hand, lots of "subseries
" (i.e. choices of specific elements from the harmonic series) do converge
, like 1+1/2+1/4+1/8+..+1/2n
+... and 1+1/4+1/9+...+1/n2
So for a sequence of natural numbers, whether the sum of their reciprocals converges or not is some measure of their density. What about 1/2+1/3+1/5+1/7+1/11+...+1/p+..., where p ranges over all the prime numbers?
By the prime number theorem, roughly one out of every ln(n) numbers near n are prime, so it appears this should diverge. But that's a big theorem, and even the deduction I outline is quite hard to get right.
There's a fairly elementary proof that the sum of the reciprocals of the primes diverges. Erdos' proof that the sum of the reciprocals of the primes diverges takes a different tack, and is completely elementary; it does seem somewhat harder to generalise, though, in the directions given below.
But just the fact of divergence is very nice, if only for the following factoid: a sum cannot diverge unless it contains infinitely many elements! So we've given a "noneuclidian" proof that there are infinitely many prime numbers! (Of course, Euclid's proof is much easier than this one)
More complex proofs can show that for any a,b with gcd(a,b)=1, the sum of all 1/p's for p prime satisfying p=a (mod b) also diverges. In particular, this proves that there are infinitely many of them -- a theorem of Dirichelet.
Another direction in which this theorem is interesting is Brun's constant.