MATH 2B Chapter Notes - Chapter 11.5: Alternating Series Test, Monotone Convergence Theorem, Subsequence

• School: UC-Irvine
• Department: Mathematics
• Course: MATH 2B
• Professor: George Donaldson

Document Summary

An alternating sequence is a sequence whose terms alternate between positive and negative. Often such sequences are written in the form an = ( 1)nbn or an = ( 1)n+1bn where (bn) is a sequence of positive terms, although sometimes they are somewhat disguised. An alternating series is the sum of an alternating sequence. The alternating series test is a convergence test which may be applied to alternating series. Suppose that (bn) is a decreasing sequence of positive values with limit zero. Like the other series tests, it does not matter which value of n denotes the initial term. As long as a series is alternating and decreasing, then it will converge. Just make sure that you observe all these facts when using the alternating series test. Examples: the alternating harmonic series n=1 ( 1)n+1 n is certainly alternating, and the sequence ( 1 n ) de- creases with limit zero.