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
a. Does the series converge?
b. What are the terms a1000 and a1001? Calculate the ratio a1001/ a1000 .
c. Compute a1000001/ a1000000
![The image presents a mathematical expression, specifically an infinite series. The series is written in summation notation as follows:
\[ \sum_{n=1}^{\infty} \frac{n}{2^n} \]
This series represents the sum of the terms \(\frac{n}{2^n}\) starting from \(n = 1\) and continuing to infinity (\( \infty \)).
### Explanation:
- **Sigma Notation (\( \sum \))**:
Sigma (\( \sum \)) denotes the sum of a sequence of terms. Here, it is used to represent the infinite sum of the given expression.
- **Index of Summation (\( n \))**:
The index of summation is \( n \), which takes integer values starting from 1 and increases without bound.
- **Upper and Lower Limits**:
The summation starts at \( n = 1 \) and continues to infinity (\( \infty \)).
- **Term of the Series (\( \frac{n}{2^n} \))**:
Each term in the series is given by the fraction \(\frac{n}{2^n}\). This represents the integer \( n \) divided by \( 2 \) raised to the power of \( n \).
The infinite series \(\sum_{n=1}^{\infty} \frac{n}{2^n}\) is an example of a geometric series with varying numerators. Understanding the convergence and sum of such series is essential in higher mathematics, particularly in the study of sequences and series in calculus.
---
On an educational website, this explanation helps students and learners comprehend the components and significance of the infinite series presented in the image. It breaks down the notation and provides a clear understanding of what the series represents.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd04de2b6-c3ee-4b26-bb77-15432c184cae%2F8ebd5a42-5cc5-4487-889e-53b200396b4d%2Fao2fb8_processed.png&w=3840&q=75)
Transcribed Image Text:The image presents a mathematical expression, specifically an infinite series. The series is written in summation notation as follows:
\[ \sum_{n=1}^{\infty} \frac{n}{2^n} \]
This series represents the sum of the terms \(\frac{n}{2^n}\) starting from \(n = 1\) and continuing to infinity (\( \infty \)).
### Explanation:
- **Sigma Notation (\( \sum \))**:
Sigma (\( \sum \)) denotes the sum of a sequence of terms. Here, it is used to represent the infinite sum of the given expression.
- **Index of Summation (\( n \))**:
The index of summation is \( n \), which takes integer values starting from 1 and increases without bound.
- **Upper and Lower Limits**:
The summation starts at \( n = 1 \) and continues to infinity (\( \infty \)).
- **Term of the Series (\( \frac{n}{2^n} \))**:
Each term in the series is given by the fraction \(\frac{n}{2^n}\). This represents the integer \( n \) divided by \( 2 \) raised to the power of \( n \).
The infinite series \(\sum_{n=1}^{\infty} \frac{n}{2^n}\) is an example of a geometric series with varying numerators. Understanding the convergence and sum of such series is essential in higher mathematics, particularly in the study of sequences and series in calculus.
---
On an educational website, this explanation helps students and learners comprehend the components and significance of the infinite series presented in the image. It breaks down the notation and provides a clear understanding of what the series represents.
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 5 steps with 8 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