Chapter 7: Q. 15 (page 656)

The Comparison Test: Let ${鈭�}_{k = 1}^{鈭�} a_{k} and {鈭�}_{k = 1}^{鈭�} b_{k}$ be two series with _ terms such that $0_a_{k}___b_{k}$ for every positive integer k. If the series ${鈭�}_{k = 1}^{鈭�} b_{k}$ ___ , then the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ ___.

Short Answer

Expert verified

Let ${鈭�}_{k = 1}^{鈭�} a_{k} and {鈭�}_{k = 1}^{鈭�} b_{k}$ be two series with positive terms such that $0 鈮� a_{k} 鈮� b_{k}$ for every positive integer k. If the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ diverges, then the series ${鈭�}_{k = 1}^{鈭�} b_{k}$ diverges.

Step by step solution

Step 1. Given Information.

The given test is the comparison test.

Step 2. Explanation

A comparison test is used if there are 2 positive series such that $0 鈮� a_{k} 鈮� b_{k}$ .
If such series are found, the if ${鈭�}_{k = 1}^{鈭�} a_{k}$ diverges, ${鈭�}_{k = 1}^{鈭�} b_{k}$ also diverges.
And if ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges, ${鈭�}_{k = 1}^{鈭�} a_{k}$ also converges.

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The Comparison Test: Let ${鈭�}_{k = 1}^{鈭�} a_{k} and {鈭�}_{k = 1}^{鈭�} b_{k}$ be two series with _ terms such that $0_a_{k}___b_{k}$ for every positive integer k. If the series ${鈭�}_{k = 1}^{鈭�} b_{k}$ ___ , then the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ ___.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation

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The Comparison Test: Let 鈭�k=1鈭�akand鈭�k=1鈭�bkbe two series with ___ terms such that 0___ak___bkfor every positive integer k. If the series 鈭�k=1鈭�bk___ , then the series 鈭�k=1鈭�ak___.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Logic and Functions

Probability and Statistics

Calculus

Geometry

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

The Comparison Test: Let ${鈭�}_{k = 1}^{鈭�} a_{k} and {鈭�}_{k = 1}^{鈭�} b_{k}$ be two series with _ terms such that $0_a_{k}___b_{k}$ for every positive integer k. If the series ${鈭�}_{k = 1}^{鈭�} b_{k}$ ___ , then the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ ___.