2 8. For the telescoping series find a formula for the nth term of the sequence of (2k-1)(2k + 1)' partial sums {Sn}. Then evaluate lim Sn to obtain the value of the series or state that the series diverges. 818
2 8. For the telescoping series find a formula for the nth term of the sequence of (2k-1)(2k + 1)' partial sums {Sn}. Then evaluate lim Sn to obtain the value of the series or state that the series diverges. 818
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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![### Problem Statement
#### Telescoping Series Analysis
8. For the telescoping series
\[ \sum_{k=1}^{\infty} \frac{2}{(2k-1)(2k+1)} \]
find a formula for the \(n\)th term of the sequence of partial sums \(\{S_n\}\). Then evaluate
\[ \lim_{n \to \infty} S_n \]
to obtain the value of the series or state that the series diverges.
---
### Explanation
**Telescoping Series**: A telescoping series is one where most terms cancel out with subsequent terms. This allows for simplification when evaluating the series.
---
**Step-by-Step Solution**:
1. **Identify Partial Fractions**:
- Decompose the given term so it can be separated into partial fractions.
2. **Find Partial Sums**:
- Derive the formula for the nth partial sum \(S_n\).
3. **Evaluate the Limit**:
- Compute the limit \(\lim_{n \to \infty} S_n\) to determine the value of the series.
4. **Conclusion**:
- Based on the limit, conclude whether the series converges or diverges.
---
**Why Telescoping Series are Important**:
- Telescoping series can simplify complex problems.
- Understanding this concept is critical for advanced calculus and mathematical analysis.
Refer to textbooks or educational resources for more detailed steps and examples on solving telescoping series problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fff7f42fd-3734-43a2-a946-a65fb76095a7%2Ff3fe1967-7b23-4a4c-bfc4-aca9bfcf5e95%2F5wff74k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
#### Telescoping Series Analysis
8. For the telescoping series
\[ \sum_{k=1}^{\infty} \frac{2}{(2k-1)(2k+1)} \]
find a formula for the \(n\)th term of the sequence of partial sums \(\{S_n\}\). Then evaluate
\[ \lim_{n \to \infty} S_n \]
to obtain the value of the series or state that the series diverges.
---
### Explanation
**Telescoping Series**: A telescoping series is one where most terms cancel out with subsequent terms. This allows for simplification when evaluating the series.
---
**Step-by-Step Solution**:
1. **Identify Partial Fractions**:
- Decompose the given term so it can be separated into partial fractions.
2. **Find Partial Sums**:
- Derive the formula for the nth partial sum \(S_n\).
3. **Evaluate the Limit**:
- Compute the limit \(\lim_{n \to \infty} S_n\) to determine the value of the series.
4. **Conclusion**:
- Based on the limit, conclude whether the series converges or diverges.
---
**Why Telescoping Series are Important**:
- Telescoping series can simplify complex problems.
- Understanding this concept is critical for advanced calculus and mathematical analysis.
Refer to textbooks or educational resources for more detailed steps and examples on solving telescoping series problems.
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