for every $n$ (it alternates signs) (2) $|a_{n+1}|\leq |a_n|$ (3) $\lim_{n\rightarrow \infty} a_n =0$ Then $\sum_{n=1}^\infty a_n$ converges. This then brings up the topic of Conditional and Absolute convergence. For a generalization of the alternating series test, see Dirichlets Test . (This test allows us to give the conditions of convergence