How do you use the Alternating Series Test?

1 Answer
Wataru
Oct 2, 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 apply the test to the alternating series below.

#sum_{n=1}^infty(-1)^{n-1}1/sqrt{n}#

In this series, #b_n=1/sqrt{n}#.

Let us check the two conditions.

  1. #1/sqrt{n} ge 1/sqrt{n+1}# for all #n ge 1#
  2. #lim_{n to infty}1/n=1/infty=0#

Hence, we conclude that the alternating series converges.

I hope that this was helpful.

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