how does $\sum_{n=0}^{\infty} (-1)^n \frac{1}{1+n}$ diverge? I thought that to prove an alternating series two tests needed to be proven
$$a_n \ge a_{n+1}$$
which is true and
$$ \lim_{n\to\infty} b_n  = 0 \ \  \ \ \ \ \text{which}  \ \ \ \ \ \  \lim_{n\to\infty}\frac{1}{1+n}=0$$
yet sources (wolfram alpha) indicate that it does not converge

In fact it converges. Let $b_n=(-1)^n$ and $a_n=\frac {1}{n+1}$. Then $a_n>0$ is decreasing that goes to $0$.Also $b_n$ has bounded partial sums because $\sum_{k=0}^{n} b_n\leq 1$.So using Dirichlet's Proposition we have that it converges. See here http://en.wikipedia.org/wiki/Dirichlet's_test
