10.7 Alternating Series Test for Convergence

An alternating series has terms that switch sign, usually because of a factor like ${(-1)^n or (-1)^{n+1}}$. The alternating series test says that a series ${\sum (-1)^n b_n}$ or ${\sum (-1)^{n+1} b_n}$ converges if the positive terms b_n satisfy two conditions: b_n decreases eventually, and ${\lim_{n\to\infty} b_n=0}$. This test is important because some alternating series converge even though the corresponding positive-term series diverges.

You should first separate the sign pattern from the size of the terms. Then check whether the absolute values decrease and go to 0. If both conditions hold, the series converges by the alternating series test. This test does not tell you absolute convergence by itself; it only establishes convergence of the alternating series. That distinction becomes important in the next topic.