Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![## Problem Statement
Find the sum of the convergent series:
\[
\sum_{n=0}^{\infty} 6 \left( \frac{6}{7} \right)^n
\]
### Explanation
This is an infinite geometric series. The series can be expressed in the form:
\[
\sum_{n=0}^{\infty} ar^n
\]
where \( a = 6 \) is the first term and \( r = \frac{6}{7} \) is the common ratio. For a geometric series to converge, the absolute value of the common ratio \( |r| \) should be less than 1. In this case, \( |r| = \frac{6}{7} < 1 \), so the series converges.
#### Sum of the Series
The sum \( S \) of an infinite geometric series is given by the formula:
\[
S = \frac{a}{1 - r}
\]
Substituting the values:
\[
S = \frac{6}{1 - \frac{6}{7}} = \frac{6}{\frac{1}{7}} = 6 \times 7 = 42
\]
Therefore, the sum of the series is 42.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45235a53-69d6-46fa-83a8-de0bf8bf66d1%2F86b8d2b4-53e1-4126-955f-d773d6e8b434%2Fo1eop5d_processed.png&w=3840&q=75)
Transcribed Image Text:## Problem Statement
Find the sum of the convergent series:
\[
\sum_{n=0}^{\infty} 6 \left( \frac{6}{7} \right)^n
\]
### Explanation
This is an infinite geometric series. The series can be expressed in the form:
\[
\sum_{n=0}^{\infty} ar^n
\]
where \( a = 6 \) is the first term and \( r = \frac{6}{7} \) is the common ratio. For a geometric series to converge, the absolute value of the common ratio \( |r| \) should be less than 1. In this case, \( |r| = \frac{6}{7} < 1 \), so the series converges.
#### Sum of the Series
The sum \( S \) of an infinite geometric series is given by the formula:
\[
S = \frac{a}{1 - r}
\]
Substituting the values:
\[
S = \frac{6}{1 - \frac{6}{7}} = \frac{6}{\frac{1}{7}} = 6 \times 7 = 42
\]
Therefore, the sum of the series is 42.
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