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"?
Show that $a_n=\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$ would not contain a natural number for all n
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Try Induction to prove it. – KON3 May 20 '15 at 09:30
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What would be the induction step? – 3SAT May 20 '15 at 09:33
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This is definitely a duplicate of another MSE question, unfortunately I can't remember the link right now – Ewan Delanoy May 20 '15 at 09:34
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There are many duplicates of this question, e.g. see here: http://math.stackexchange.com/questions/24113/prove-that-1-frac12-frac13-frac1n-is-not-an-integer – Dietrich Burde May 20 '15 at 09:35
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1@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 09:41
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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.
Augustin
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