2
$\begingroup$

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"?

$\endgroup$

marked as duplicate by Daniel Fischer May 20 '15 at 9:34

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Try Induction to prove it. $\endgroup$ – Anjan3 May 20 '15 at 9:30
  • $\begingroup$ What would be the induction step? $\endgroup$ – 3SAT May 20 '15 at 9:33
  • $\begingroup$ This is definitely a duplicate of another MSE question, unfortunately I can't remember the link right now $\endgroup$ – Ewan Delanoy May 20 '15 at 9:34
  • $\begingroup$ There are many duplicates of this question, e.g. see here: math.stackexchange.com/questions/24113/… $\endgroup$ – Dietrich Burde May 20 '15 at 9:35
  • 1
    $\begingroup$ @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. $\endgroup$ – Daniel Fischer May 20 '15 at 9:41
2
$\begingroup$

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.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.