Post Reply 
 
Thread Rating:
  • 0 Votes - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
For \(n > 1\), $s = 1+1/2+\cdots+1/n$ is not an integer
08-04-2010, 02:44 AM
Post: #1
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. $\square$
Find all posts by this user
Quote this message in a reply
Post Reply 


Forum Jump:


Contact Us | Software Frontier | Return to Top | Return to Content | Lite (Archive) Mode | RSS Syndication