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Chapter 9: Problem 4

Show that the Alternating Series Test is a consequence of Dirichlet's Test \(9.3 .4 .\)

Short Answer

Expert verified
The conditions required for an alternating series to converge, as per the Alternating Series Test, fall directly under the stipulations of Dirichlet's Test. Therefore, one can conclude that the Alternating Series Test is a consequence of Dirichlet's Test.

Step by step solution

01

Understand the Alternating Series Test

The Alternating Series Test states that for alternating series of the form \( \sum_{n=1}^{\infty} (-1)^{n+1} b_n \) where \( b_n \) denotes the terms of the series and are all positive, if the following two conditions are satisfied: (1) \( b_{n+1} \leq b_n \) for every \( n \in \mathbb{N} \) i.e., the sequence \( \{b_n\} \) is monotonically decreasing, and (2) \( \lim_{n \to \infty} b_n = 0 \) i.e., the sequence \( \{b_n\} \) converges to 0, then the alternating series converges.
02

Understand Dirichlet's Test

Dirichlet's Test states that for any two sequences \( \{a_n\}\) and \( \{b_n\}\), if the following two conditions are satisfied: (1) \( \{a_n\} \) is a sequence of real numbers such that the partial sums \( A_n = \sum_{k=1}^{n} a_k \) are bounded. That is, there is a positive number \( M \) such that \( |A_n| \leq M \) for all \( n \in \mathbb{N} \); and (2) \( \{b_n\} \) is a monotonically decreasing sequence of positive numbers that converges to 0, then the series \( \sum_{n=1}^{\infty} a_n b_n \) converges.
03

Apply Dirichlet's Test to Alternating Series

We need to show that the Alternating Series Test is a consequence of Dirichlet's Test. Consider an alternating series \( \sum_{n=1}^{\infty} (-1)^{n+1} b_n \). Let \( a_n = (-1)^{n+1} \) and \( b_n = b_n \). Since \( \{b_n\} \) is monotonically decreasing and converges to 0, it satisfies condition 2 of Dirichlet's Test. Next, the sequence \( \{a_n\} \) is a alternating sequence of 1 and -1, so its partial sums are bounded by 1, satisfying condition 1 of Dirichlet's Test. Therefore, by Dirichlet's Test, the alternating series converges. Hence, the Alternating Series Test is a direct consequence of Dirichlet's Test.

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