What is the Alternating Series Test of convergence?

Question

Elijah Abram · Accepted Answer

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.

Cameron Alexander · Answer

undefined The Alternating Series Test of convergence is a method used to determine whether an alternating series converges or diverges. It states that if the terms of an alternating series decrease in absolute value and approach zero, and if the series satisfies the conditions of Leibniz's theorem, then the series converges. Leibniz's theorem requires that the terms alternate in sign and that the absolute values of the terms decrease monotonically to zero. If these conditions are met, the alternating series is said to converge.