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
![**Transcription:**
Determine whether
\[
\sum_{{n=1}}^{\infty} (-1)^{n-1} \frac{1}{\sqrt{n} + 3}
\]
is convergent.
---
**Explanation:**
This mathematical expression is an infinite series where each term of the series is given by
\[
(-1)^{n-1} \frac{1}{\sqrt{n} + 3}
\]
- **Summation Notation (\(\sum\)):** This symbol indicates that the terms following it are to be added together from \(n=1\) to infinity (\(\infty\)).
- **\((-1)^{n-1}\):** This factor alternates the sign of each term in the series, making it an alternating series.
- **\(\frac{1}{\sqrt{n} + 3}\):** Each term consists of the fraction of 1 divided by the quantity \((\sqrt{n} + 3)\), where \(\sqrt{n}\) is the square root of \(n\).
To determine whether this series is convergent, you would typically apply convergence tests specifically suited for alternating series, such as the Alternating Series Test, and possibly consider other tests for convergence to deepen your analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb383ffc7-8622-4111-9f80-ade3985dd461%2F89739fcb-ac6e-4500-956a-fe8e1a5a9131%2Fgxbmun_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription:**
Determine whether
\[
\sum_{{n=1}}^{\infty} (-1)^{n-1} \frac{1}{\sqrt{n} + 3}
\]
is convergent.
---
**Explanation:**
This mathematical expression is an infinite series where each term of the series is given by
\[
(-1)^{n-1} \frac{1}{\sqrt{n} + 3}
\]
- **Summation Notation (\(\sum\)):** This symbol indicates that the terms following it are to be added together from \(n=1\) to infinity (\(\infty\)).
- **\((-1)^{n-1}\):** This factor alternates the sign of each term in the series, making it an alternating series.
- **\(\frac{1}{\sqrt{n} + 3}\):** Each term consists of the fraction of 1 divided by the quantity \((\sqrt{n} + 3)\), where \(\sqrt{n}\) is the square root of \(n\).
To determine whether this series is convergent, you would typically apply convergence tests specifically suited for alternating series, such as the Alternating Series Test, and possibly consider other tests for convergence to deepen your analysis.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 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