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Skinner · #12 January 31st, 2008, 10:14 PM

Let me have a try:

problem:

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

Solution:

If you can prove that $(\frac{1}{2} + ... + \frac{1}{n})$ is not an integer, then you can prove that $1+\frac{1}{2}+...+\frac{1}{n}$ is not an integer.

Why? if you add 1 to anything that's not an integer, you get a non-integer value.

look at this function:

$f(n) = \frac{1}{n} + \frac{1}{n-1} + \frac{1}{n-2} + ... + \frac{1}{2}$

This rational function has asymptopes at the integers, which means that at every integer, the function approaches it, but never reaches it. This can be proven if we took the limit of the function as $x \rightarrow 1+, 2+, ....$ and so on. This means that the value of $f(n)$ never becomes an integer.

I conclude that $f(n) = \frac{1}{n} + \frac{1}{n-1} + \frac{1}{n-2} + ... + \frac{1}{2} = (\frac{1}{2} + ... + \frac{1}{n})$

$1 + f(n)$ is not an integer, therefore $1+\frac{1}{2}+...+\frac{1}{n}$ is not an integer either.

Please tell me what you think.

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