# Show that $a_n=\frac{1}{2}+\frac{1}{3}+…+\frac{1}{n}$ would not contain a natural number for all n [duplicate]

Show that the series: $a_n=\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$ would not contain natural number for all n

Can I prove that using "simple tools"?

## marked as duplicate by Daniel FischerMay 20 '15 at 9:34

• @Nehorai The sums differ by $\frac{1}{1}$, and $x$ is an integer if and only if $x+1$ is one. The interesting part of the problem is exactly the same in both questions. – Daniel Fischer May 20 '15 at 9:41
Yes you can prove this using simple tools. For example you can prove by induction that $\forall n\in\mathbb{N}$, $S_n$ is a ratio of an odd number and an even number, so it's not an integer.