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.