What is the Alternating Series Test of convergence?

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What is the Alternating Series Test of convergence?

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Answer 1 · 2014-09-12T10:58:55

Alternating Series Test states that an alternating series of the form
$sum_{n=1}^infty (-1)^nb_n$ , where $b_n ge0$ ,
converges if the following two conditions are satisfied:
1. $b_n ge b_{n+1}$ for all $n ge N$ , where $N$ is some natural number.
2. $lim_{n to infty}b_n=0$

Let us look at the alternating harmonic series $sum_{n=1}^infty (-1)^{n-1}1/n$ .
In this series, $b_n=1/n$ . Let us check the two conditions.
1. $1/n ge 1/{n+1}$ for all $n ge 1$
2. $lim_{n to infty}1/n=0$

Hence, we conclude that the alternating harmonic series converges.

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What is the Alternating Series Test of convergence?

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