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DivideBy0 · #9 January 26th, 2008, 06:08 AM

Quote:

Originally Posted by ThePerfectHacker

1)Let $n\geq 2$ prove that $1+\frac{1}{2}+...+\frac{1}{n}$ is not an integer.

For n = 2, $1+\frac{1}{2}= \frac{3}{2}$ is not an integer.

Assume $1+\frac{1}{2}+...+\frac{1}{n}$ is not an integer for $2 \leq n \leq k$ .

Then, take $n = k$ , let

$1+\frac{1}{2}+...+\frac{1}{k}=\frac{k!+\frac{k!}{2}+...+(k-1)!}{k!}=\frac{\sum_{x=1}^k \frac{k!}{x}}{k!}$

For this to not be an integer, we must have $k!\ \not|\ \sum_{x=1}^k \frac{k!}{x}$

So we must have

$\sum_{x=1}^k \frac{k!}{x}\neq ak!$ for some integer $a$ . (and of course $k \geq 2$ )

$k! \sum_{x=1}^k \frac{1}{x} \neq ak!$

$\sum_{x=1}^k \frac{1}{x} \neq a$

NOOOOOOOOOOOOOO!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!1