Consider the following alternating series
(where a_{n} > 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. a_{n} ≥ a_{n +1} for all n ≥ N, where N ≥ 1, and
2. \(\mathop {\lim }\limits_{n \to \infty } {a_n} = 0\)
