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Suzan · #14 October 11th, 2008, 12:49 AM

Quote:

Originally Posted by ThePerfectHacker

1) Let $n\geq 2$ prove that $1 - \frac{1}{2}+\frac{1}{3} - ... \pm \frac{1}{n}$ is not an integer.

Suppose that

. Choose an integer

such that

.
Then

Consider the lowest common multiple of

. This number will be of the form

, where

is an odd integer. Now multiply both sides of the equation by this number, to get

Now, when multiplied out, all the terms on the left will be integers, except one:

is not an integer, since

is odd. So the left hand side is not an integer, and hence neither is the right hand side. That means that

is not an integer.

Not:
http://plus.maths.org/issue12/features/harmonic/index.html