Expii

# Alternating Series Test - Expii

An alternating series $$\sum a_n$$ is a series whose terms alternate from positive to negative. The alternating series test says that an alternating series definitely converges if $$|a_n| \geq |a_{n+1}|$$ for all $$n$$, and $$\lim_{n\to\infty} a_n = 0$$. (For example, the alternating harmonic series, $$\frac11 - \frac12 + \frac13 - \frac14 \pm \dots$$, converges, even though the harmonic series diverges.) In other words, the test applies to alternating series in which the summands $$a_n$$ are "decreasing in absolute value and converging to $$0$$ as $$n\to\infty$$". However, the test is inconclusive otherwise.