In this section we will learn about what alternating series are. It is very easy to see if the series is alternating by expanding it out. If the terms go from positive to negative and negative to positive, then it is alternating. We will also examine the convergence of alternating series by using a method called the alternating series test. The test requires two conditions, which is listed below. Keep in mind that if you cannot fulfill these conditions, that does not mean the alternating series is divergent. There is still a possibility that it is convergent.
An Alternating series is in the form: ∑(−1)nbn
Where bn≥0 An alternating series is not limited to these two forms because the exponent on the (-1) can vary.
The Alternating Series Test states that if the two following conditions are met, then the alternating series is convergent:
1. limn →∞bn=0
2. The sequence bn is a decreasing sequence.
For the second condition, bn does not have to be strictly decreasing for all n≥1. As long as the sequence is decreasing for n→∞, then that will be sufficient enough.
Convergence of the Alternating Series Test
Show that the following series converge:
Alternating series test
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