Let's consider the serie $$\sum_{k=1}^n \frac{1}{k} = \frac{s_n}{t_n}$$ where $ s_n$ and $t_n $ are coprime. Is there a way to find an inductive relation between $ s_n,t_n, s_{n+1}, t_{n+1}$?
We know that $$ S = \sum_{k=1}^{n+1} \frac{1}{k} - \sum_{k=1}^n \frac{1}{k} = \frac{(n+1)s_n+t_n}{t_n(n+1)} $$ The only problem left is to show that the numerator and denominator of this expression $S$ are coprime.