Proving Sum of Reciprocal of Natural Numbers is Not an Integer

[Dragonfall]

How do I show that $$\sum_1^n\frac{1}{k}$$ is not an integer for n>1? I tried bounding them between two integrals but that doesn't cut it. I know that $$\sum_1^n\frac{1}{k}=\frac{(n-1)!+n(n-2)!+n(n-1)(n-3)!+...+n!}{n!}$$ but I can't get a contradiction.

 

[StatusX]

https://www.physicsforums.com/showthread.php?t=54831"

Here's an old thread with a similar problem. The third post has a nice hint.