This number is not natural

Question

Let $k$ be a natural number, such that $k>1$.

Show that $1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{k}$ is not a natural number.

How I can prove this?

Have you looked up on the web ? – Shailesh Aug 21 '16 at 01:13 — Shailesh, Aug 21 '16 at 01:13
this question has been asked a many many times before – Asinomás Aug 21 '16 at 01:59 — Asinomás, Aug 21 '16 at 01:59

ajotatxe · Answer 1 · 2016-08-21T01:29:07.103

1

Hints:

Consider the highest power of $2$ that divides $k$, and the highest power of $2$ lesser than $k$.
Bertrand's postulate leads to a shorter solution. This result is highly non-trivial, though, and it is often not allowed in introductory courses of number theory.

edited Aug 21 '16 at 01:29

answered Aug 21 '16 at 01:21

ajotatxe

1 Answers1