Is there an elementary proof that sum $1/3+1/5+1/7+...+1/(2n+1)$ is not an integer for every n?
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$\begingroup$ hint: chebyshev's theorem. $\endgroup$– Yiyi RaoCommented Nov 18, 2016 at 13:53
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1$\begingroup$ Possible duplicate of Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer? $\endgroup$– Dietrich BurdeCommented Nov 18, 2016 at 16:00
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We have shown here that $$\begin{align*} \sum_{k=1}^n \frac1{2k-1}&=\frac12(2H_{2n}-H_n), \end{align*}$$ where $H_n$ is the harmonic number. Now $H_n$ for $n>1$ is never an integer, see this MSE-question, where several elementary proofs have been given. This implies that no difference $H_{n+k}-H_n$ is an integer, see here. The same reasoning should work to see that also $H_{2n}-H_n/2$ is never an integer.
answered Nov 18, 2016 at 15:57
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$\begingroup$ But $2H_{2n}-H_n$ could be, no ? $\endgroup$– user65203Commented Nov 18, 2016 at 16:07
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$\begingroup$ No, but good point. I have to fill the gap.. $\endgroup$ Commented Nov 18, 2016 at 16:09