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Aryth · #4 April 25th, 2008, 02:31 PM

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.

The series you presented is the Alternating Harmonic Series, which is Conditionally Convergent, the series is represented by:

$\sum_{n=1}^{\infty} \left(\frac{(-1)^{n+1}}{n}\right)$

The series' terms look like such:

$1 - \frac{1}{2} + \frac{1}{3} - ... \pm \frac{1}{n}$

This series converges to $\ln{2}$

Since the series converges to $\ln{2}$ and since:

$|a_{n+1}| < |a_n|$

Then for $n \geq 2$ the series can never reach one since it is incrementing up or down by smaller amounts. Since you subtract $\frac{1}{2}$ from 1 for n=2, and since the terms are decreasing and alternating in sign, then the series will never reach one again, therefore, this can't be an integer for $n \geq 2$ because all terms are decreasing,therefore the partial sums remain between 1 and 0.

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