Question

Why does this series converge by the alternating series test?

Answer 1

When you’re talking about convergence as n approaches infinity, you really only care about the end behavior of the series. In this case, the series is decreasing from n = 2 onwards.

Answer 2

If the question really asked about _absolute convergence_ then the alternating series test isn't the right tool - that simply gives you convergence. But we actually do have absolute convergence here as well, e.g. via the comparison test, since |(-1)^n ln(n)/n!| ≤ 1/(n-1)! for all sufficiently large n.