Advanced P9(BOMC 2)

[*Mahi*]

Prove that the sum \[S(m,n) = \frac 1m + \frac 1{m+1} + \cdots \frac 1{m+n}\]
Is not an integer for any given positive integers $m$ and $n$.

[Phlembac Adib Hasan]

Hint:
Firstly,

Sum up the fractions.

Next step:

Bertrand's postulate says there is a prime between $\frac {m+n}{2}$ and $m+n$ (for even) and between $\frac {m+n-1}{2}$ and $m+n-1$(for odd).
(Also notice this trick does not work for $m>n$.So how can we use Bertrand's postulate there?Just simple.You have to find a different value for $k$ and $\frac {k}{2}$.Find it and see how the problem solves itself.)