What is the Alternating Series Test of convergence?

1 Answer
Wataru
Sep 12, 2014

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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