6. Find A and B such that for all i 2 2 A B %3D 2 - 1 i-1 i+1' | and use this identity to express the partial sum S, = E2 2/(i² – 1) of the series Σ n2 - 1 n22
6. Find A and B such that for all i 2 2 A B %3D 2 - 1 i-1 i+1' | and use this identity to express the partial sum S, = E2 2/(i² – 1) of the series Σ n2 - 1 n22
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
![**Problem 6**
Find \( A \) and \( B \) such that for all \( i \geq 2 \),
\[
\frac{2}{i^2 - 1} = \frac{A}{i - 1} + \frac{B}{i + 1}
\]
and use this identity to express the partial sum \( s_n = \sum_{i=2}^{n} \frac{2}{i^2 - 1} \) of the series
\[
\sum_{n \geq 2} \frac{2}{n^2 - 1}
\]
as a telescoping sum. Use this telescoping sum to show that the series is convergent, and find its sum.
---
**Explanation of Approach:**
The problem involves finding constants \( A \) and \( B \) such that the decomposition of the fraction provides a way to simplify the summation into a telescoping series. A telescoping series is one where most terms cancel out, leaving only a few terms that are easy to evaluate.
**To solve the problem:**
1. **Decompose \( \frac{2}{i^2 - 1} \):**
- Recognize that \( i^2 - 1 = (i-1)(i+1) \).
- Set up the partial fraction: \( \frac{2}{i^2 - 1} = \frac{A}{i-1} + \frac{B}{i+1} \).
- Determine \( A \) and \( B \) such that this identity holds for all \( i \geq 2 \).
2. **Find \( s_n \):**
- Express the partial sum \( s_n = \sum_{i=2}^{n} \frac{2}{i^2 - 1} \) using the identity.
- Identify the cancellation pattern in the series to simplify the summation.
3. **Analyze Convergence:**
- Use the nature of the telescoping series to argue the convergence of the series.
- Calculate the sum of the convergent series.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0ec25ffb-8aef-468b-ad53-a5caf8bbc7e2%2Fa4939e08-ab3e-4e9b-963b-8f863872219a%2F678tw1l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 6**
Find \( A \) and \( B \) such that for all \( i \geq 2 \),
\[
\frac{2}{i^2 - 1} = \frac{A}{i - 1} + \frac{B}{i + 1}
\]
and use this identity to express the partial sum \( s_n = \sum_{i=2}^{n} \frac{2}{i^2 - 1} \) of the series
\[
\sum_{n \geq 2} \frac{2}{n^2 - 1}
\]
as a telescoping sum. Use this telescoping sum to show that the series is convergent, and find its sum.
---
**Explanation of Approach:**
The problem involves finding constants \( A \) and \( B \) such that the decomposition of the fraction provides a way to simplify the summation into a telescoping series. A telescoping series is one where most terms cancel out, leaving only a few terms that are easy to evaluate.
**To solve the problem:**
1. **Decompose \( \frac{2}{i^2 - 1} \):**
- Recognize that \( i^2 - 1 = (i-1)(i+1) \).
- Set up the partial fraction: \( \frac{2}{i^2 - 1} = \frac{A}{i-1} + \frac{B}{i+1} \).
- Determine \( A \) and \( B \) such that this identity holds for all \( i \geq 2 \).
2. **Find \( s_n \):**
- Express the partial sum \( s_n = \sum_{i=2}^{n} \frac{2}{i^2 - 1} \) using the identity.
- Identify the cancellation pattern in the series to simplify the summation.
3. **Analyze Convergence:**
- Use the nature of the telescoping series to argue the convergence of the series.
- Calculate the sum of the convergent series.
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 1 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