Why is sigma (-1)^n/n a convergent series while sigma 1/n is a divergent p-series?
...is it because half the terms are used for either sign +/- and therefore each set decays more rapidly than 1/n?
Let $\displaystyle a_n=\frac1n,$ then obviously $\displaystyle \lim_{n\to\infty}\frac1n=0.$ On the other hand, $\displaystyle a_n$ is a strictly decreasing sequence for $\displaystyle n\ge1.$ Hence, by Leibniz test the series converges.
But, this series converges conditionally, since $\displaystyle \left| a_{n} \right|=\frac{1}{n}$ and $\displaystyle \sum\limits_{n=1}^{\infty }{\frac{1}{n}}$ diverges.