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 For $$n > 1$$, $s = 1+1/2+\cdots+1/n$ is not an integer
08-04-2010, 02:44 AM
Post: #1
 elim Moderator Posts: 578 Joined: Feb 2010 Reputation: 0
For $$n > 1$$, $s = 1+1/2+\cdots+1/n$ is not an integer
Proof. Take $k \in \mathbb{N}, \; 2^k \le n < 2_{k+1}$, and $P=\prod_{1\le m \le \lfloor (n+1)/2\rfloor}(2m-1),\;$ then except $2^{k-1}P/2^k = P/2$, all the summands in the right hand side of $2^{k-1}P\cdot s = 2^{k-1}P + 2^{k-1}P/2 + \cdots + 2^{k-1}P/n$ are integers. Therefore $s$ is not an integer. $\square$
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