Alternating Series Test - Sequences and Series

The final two tests that we looked at for series convergence has needed that all the terms in the series be positive.  Actually there are several series out there that have negative terms in them and thus we now need to start looking at tests for these types of series.

The test which we are going to look into in this part will be a test for alternating series.  An alternating series is some series, ∑a_n , for that the series terms can be written in one of the subsequent two forms.

a_n = (-1)ⁿ b_n                                       b_n > 0

a_n = (-1)ⁿ⁺¹ b_n                                             b_n > 0

There are so many other ways to deal with the alternating sign, although they can all be written as one of the above two forms.  For illustration,

(-1)ⁿ⁺² = (-1)ⁿ (-1)² = (-1)ⁿ

(-1)ⁿ⁺¹ _­= (-1)ⁿ⁺¹ (-1)^-2 = (-1)^n+!

Certainly there are many others, but they all follow similar basic pattern of reducing to one of the first two forms that given.  If you should happen to run into a dissimilar form than the first two, don't worry about transforming it to one of those forms, just be alert that it can be and thus the test from this section can be used.