Alternating Series

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Alternating Series and the Alternating Series Test

An alternating series is a series where $a_n$ flips sign every turn. For instance,

$\sum_{n=1}^\infty \frac{(-1)^n}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac 14 + \ldots.$

Alternating Series Test: If { $b_n$ } is a non-increasing sequence of non-negative numbers with $\displaystyle{\lim_{n \to \infty} b_n = 0}$ , then $\displaystyle{\sum_{n=1}^\infty (-1)^{n-1} b_n = b_1 - b_2 + b_3 - b_4 + \ldots}$ converges.