   Chapter 11, Problem 22RQ

Chapter
Section
Textbook Problem

# Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.22. If ∑ n = 1 ∞ a n = A and ∑ n = 1 ∞ b n = B , then ∑ n = 1 ∞ a n b n = A B .

To determine

Whether the statement “If n=1an=A and n=1bn=B , then n=1anbn=AB ” is true or false.

Explanation

Counter Example:

The series n=1an=n=1(0.1)n and n=1bn=n=1(0.2)n .

Result used:

The sum of the geometric series n=1arn1 (or) a+ar+ar2+ is a1r , where a is the first term of the series and r is the common ratio of the series.

Calculation:

The series n=1an=n=1(0.1)n is geometric series with the first term of the series is a=0.1 and the common ratio r=0.1

Use the Result stated above, the sum of the series is computed as follows,

n=1(0.1)n=0.110.1=0.10.9=19

That is, n=1an=A where n=1an=n=1(0.1)n and A=19 .

The series n=1an=n=1(0.2)n is geometric series with the first term of the series is a=0

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