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...
Related questions
Question
100%
Help with the following question
The second photo is the rest of the multiple choice
![**Determine whether the following series converges. Justify your answer.**
\[
\sum_{k=1}^{\infty} \frac{8^k + 10}{8^k}
\]
---
**Options:**
- **A.** The series is a geometric series with a common ratio \(\frac{1}{8}\). This is less than 1, so the series converges by the properties of a geometric series.
- **B.**
Because \(\frac{8^k + 10}{8^k} \leq \frac{8^k}{8^k} = 1\) for any positive integer \(k\) and \(\sum_{k=1}^{\infty} \frac{8^k}{8^k} = \sum_{k=1}^{\infty} \frac{1}{1}\) converges, the given series converges by the Comparison Test.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcef3f1b2-56cd-4f45-949e-fc7a38ef994c%2F2abec71d-1aa2-46da-b3cd-cc6a417cb6cf%2F3hruv4c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determine whether the following series converges. Justify your answer.**
\[
\sum_{k=1}^{\infty} \frac{8^k + 10}{8^k}
\]
---
**Options:**
- **A.** The series is a geometric series with a common ratio \(\frac{1}{8}\). This is less than 1, so the series converges by the properties of a geometric series.
- **B.**
Because \(\frac{8^k + 10}{8^k} \leq \frac{8^k}{8^k} = 1\) for any positive integer \(k\) and \(\sum_{k=1}^{\infty} \frac{8^k}{8^k} = \sum_{k=1}^{\infty} \frac{1}{1}\) converges, the given series converges by the Comparison Test.
![**Transcription for Educational Website**
---
**Options for Series Convergence Analysis**
**C.** Because
\[
\frac{8}{k} < \frac{8}{k} + \frac{10}{k} \text{ for any positive integer } k \text{ and } \sum \frac{8}{k}
\]
diverges, the given series diverges by the Comparison Test.
**D.** The Ratio Test yields \( r = \_\_\_ \). This is less than 1, so the series converges by the Ratio Test.
*(Type an exact answer.)*
**E.** The series is a geometric series with common ratio \( \_\_\_ \). This is greater than 1, so the series diverges by the properties of a geometric series.
**Buttons Below:**
- Clear all
- Check answer
**Note:** This exercise involves applying different convergence tests (Comparison Test, Ratio Test, and properties of geometric series) to determine the behavior of series. Each option outlines a method for assessing divergence or convergence based on mathematical principles.
---
This transcription breaks down each option methodically, explaining the reasoning behind the choice of test and the expected outcome regarding series behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcef3f1b2-56cd-4f45-949e-fc7a38ef994c%2F2abec71d-1aa2-46da-b3cd-cc6a417cb6cf%2Fnoa8dul_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website**
---
**Options for Series Convergence Analysis**
**C.** Because
\[
\frac{8}{k} < \frac{8}{k} + \frac{10}{k} \text{ for any positive integer } k \text{ and } \sum \frac{8}{k}
\]
diverges, the given series diverges by the Comparison Test.
**D.** The Ratio Test yields \( r = \_\_\_ \). This is less than 1, so the series converges by the Ratio Test.
*(Type an exact answer.)*
**E.** The series is a geometric series with common ratio \( \_\_\_ \). This is greater than 1, so the series diverges by the properties of a geometric series.
**Buttons Below:**
- Clear all
- Check answer
**Note:** This exercise involves applying different convergence tests (Comparison Test, Ratio Test, and properties of geometric series) to determine the behavior of series. Each option outlines a method for assessing divergence or convergence based on mathematical principles.
---
This transcription breaks down each option methodically, explaining the reasoning behind the choice of test and the expected outcome regarding series behavior.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning