Thread: Problem 48
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Old April 21st, 2008, 03:09 PM
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Originally Posted by ThePerfectHacker View Post
1) Let n\geq 2 prove that 1 - \frac{1}{2}+\frac{1}{3} - ... \pm \frac{1}{n} is not an integer.
Let k ÎZ such that 2^k £ n < 2^k+1

Let m be the least common multiple of 1,2,3,…,n except 2^k.

Then multiplying S = 1 – 1/2 + 1/3 -…..± 1/n by m we have:

mS = m – m/2 + m/3 -……± m/n

Each number on the right hand side is an integer except m/2^k and hence Sm is not an integer, which implies Sm is not an integer.