It is well known that the harmonic sum $\sum_1^\infty \frac{1}{n}$ diverges to infinity, and more over that we can approximate using the integral of $\frac{1}{x}$ to get that $\sum_1^N \frac{1}{n} \sim \ln(n)$.
In my problem I have a fixed finite set of primes $S$, and I want to approximate $\sum_{n\in <S>, n\leq N} \frac{1}{n}$ where $<S>$ is the set of natural numbers which are product of primes from $S$. Unlike the harmonic sum, when $N$ goes to infinity this sum converges to $\prod_{p\in S}\frac{p}{p-1}$, so the question is what is the rate of convergence of this summation.