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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![### Series Evaluation in Geometric Sequences
**Problem 7**: Evaluate each geometric series or state that it diverges. Provide reasons for your conclusion.
#### (a)
\[
\sum_{k=0}^{\infty} \left( -\frac{6}{5} \right)^k
\]
#### (b)
\[
\sum_{k=0}^{\infty} \frac{5 \cdot 6^k}{7^k}
\]
_For each series, determine if it converges or diverges and justify your reasoning:_
---
### Explanation:
#### Convergence and Divergence of Geometric Series
A geometric series:
\[
\sum_{k=0}^{\infty} ar^k
\]
1. **Converges** if \( |r| < 1 \). The series sum is given by:
\[
\sum_{k=0}^{\infty} ar^k = \frac{a}{1-r}
\]
2. **Diverges** if \( |r| \geq 1 \).
#### Analysis for (a)
\[
\sum_{k=0}^{\infty} \left( -\frac{6}{5} \right)^k
\]
Here, \( a = 1 \) and \( r = -\frac{6}{5} \).
- Since \( |r| = \left| -\frac{6}{5} \right| = \frac{6}{5} > 1 \), the series diverges.
#### Analysis for (b)
\[
\sum_{k=0}^{\infty} \frac{5 \cdot 6^k}{7^k}
\]
Rewrite the given series:
\[
\sum_{k=0}^{\infty} 5 \left( \frac{6}{7} \right)^k
\]
Here, \( a = 5 \) and \( r = \frac{6}{7} \).
- Since \( |r| = \left| \frac{6}{7} \right| = \frac{6}{7} < 1 \), the series converges.
- The sum of the series is:
\[
\sum_{k=0}^{\infty} 5 \left( \frac{6}{7}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fff7f42fd-3734-43a2-a946-a65fb76095a7%2Fd2ced440-20ac-4bd9-a207-8c46daf911b2%2F2ojebl8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Series Evaluation in Geometric Sequences
**Problem 7**: Evaluate each geometric series or state that it diverges. Provide reasons for your conclusion.
#### (a)
\[
\sum_{k=0}^{\infty} \left( -\frac{6}{5} \right)^k
\]
#### (b)
\[
\sum_{k=0}^{\infty} \frac{5 \cdot 6^k}{7^k}
\]
_For each series, determine if it converges or diverges and justify your reasoning:_
---
### Explanation:
#### Convergence and Divergence of Geometric Series
A geometric series:
\[
\sum_{k=0}^{\infty} ar^k
\]
1. **Converges** if \( |r| < 1 \). The series sum is given by:
\[
\sum_{k=0}^{\infty} ar^k = \frac{a}{1-r}
\]
2. **Diverges** if \( |r| \geq 1 \).
#### Analysis for (a)
\[
\sum_{k=0}^{\infty} \left( -\frac{6}{5} \right)^k
\]
Here, \( a = 1 \) and \( r = -\frac{6}{5} \).
- Since \( |r| = \left| -\frac{6}{5} \right| = \frac{6}{5} > 1 \), the series diverges.
#### Analysis for (b)
\[
\sum_{k=0}^{\infty} \frac{5 \cdot 6^k}{7^k}
\]
Rewrite the given series:
\[
\sum_{k=0}^{\infty} 5 \left( \frac{6}{7} \right)^k
\]
Here, \( a = 5 \) and \( r = \frac{6}{7} \).
- Since \( |r| = \left| \frac{6}{7} \right| = \frac{6}{7} < 1 \), the series converges.
- The sum of the series is:
\[
\sum_{k=0}^{\infty} 5 \left( \frac{6}{7}
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