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meymathis · #11 July 7th, 2008, 10:06 PM

Ok, how about this. It is basically Aryth's post but a bit more explicit.

Let $A_N = 1 - \frac{1}{2} + \ldots \pm\frac{1}{N}$ which is just the partial sums.

Consider the (sub) sequence of partial sums:

$O_N = 1 - \frac{1}{2} + \ldots + \frac{1}{2N+1}$ for $N\geq1$

$O_N$ is a subsequence of $A_N$ which as noted above converges to ln(2) (derive using MacLauren expansion of ln at x=1). Then $O_N\rightarrow\ln(2)$ .

$O_N$ is monotonically decreasing:

$O_{N+1}-O_N = - \frac{1}{2N+2} + \frac{1}{2N+3} < 0$

Note that $O_0=1$ and so $1>O_N\geq\ln(2)\approx0.693$ for $N>0$ and so cannot be an integer.

Likewise for the partial sums:

$E_N = 1 + \ldots - \frac{1}{2N}$ for $N\geq1$

except that $E_N$ monotonically increases from 1/2 to ln(2).

Put it together and we just showed the odd and even elements of the partial sums $A_N$ are never integers after 1.