Alternating Series Test Reference
$$ \sum_{i=0}^\infty \frac{(-1)^n}{n} $$
This alternating series fails the p-series test because the exponent of n = 1.
Yet it seems to pass the alternating series test.
1 – $a_n$ must be positive. True.
2 – Terms must be decreasing. $\frac{d}{dn} 1/n = -n^{-2}$, which is < 1. True.
3 – $ \lim_{n\rightarrow\infty} 1/n = 0 $ True.
$(-1)^n/n$ is clearly a divergent series, so why does it pass the AST?
Best Answer
What you are noting is that the series $$\sum_{i=0}^\infty \,\frac{(-1)^n}{n}\quad {\bf {converges},}$$ as you found by the alternating series test, but does not converge absolutely: $$\sum_{i=0}^\infty \,\left|\frac{(-1)^n}{n}\right| \quad = \quad \sum_{i = 0}^\infty\,\frac 1n\quad\bf{does\; not\; converge.}$$
Note: the $p$-series test is applicable for sums of the form: $\displaystyle\sum \frac 1{n^{p}},$ and your "given" series does not "fit" that form for odd $n$; indeed, the most appropriate test to use here, as you used in the end, is the alternating series test.