Divergence of a series containing primes

Question

Is there an easy proof showing that the series $1/p$, where $p$ changes over prime numbers, is divergent?

Answer 1

The proof by Tom Apostol (section 1.6 in his Intro to Analytic Number Th) is clean. If it converges then we can find $k$ so that if $p_m$ is the $m$th prime then the sum from $k+1$ onwards is less than $\frac{1}{2}$. Let $Q$ the product of $p_1,\dots,p_k$. Consider the numbers $1+nQ$ for $n=1,2,\dots$. None of these is divisible by any of the first $k$ primes. Therefore all the prime factors of $1+nQ$ occur amongst the later primes. Now it is easy to see that $\frac{1}{1+nQ}$ converges. Contradiction.