index: click on a letter A B C D E F G H I J K L M N O P Q R S T U V W X Y Z A to Z index index: subject areas numbers & symbols sets, logic, proofs geometry algebra trigonometry advanced algebra & pre-calculus calculus advanced topics probability & statistics real world applications multimedia entries

Alternating Series Test

A convergence test for alternating series.

 Consider the following alternating series (where an > 0 for all n) and/or its equivalents: $\sum\limits_{k = 1}^\infty {{{\left( { - 1} \right)}^{k + 1}}{a_k}} = {a_1} - {a_2} + {a_3} - {a_4} + \cdots$ The series converges if the following conditions are met. 1. an ≥ an +1 for all n ≥ N, where N ≥ 1, and 2. $$\mathop {\lim }\limits_{n \to \infty } {a_n} = 0$$