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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Solve 79 and 81 please also use the sequence of partial sums to determine whether the series converges or diverges
![Certainly! Here’s a transcription of the text as it might appear on an educational website:
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### Series and Convergence
**78.** \( S_n = 2 - (n + 2)/2^n, n \geq 1 \)
For each of the following series, use the **sequence of partial sums** to determine whether the **series converges or diverges**.
**79.** \[ \sum_{n=1}^{\infty} \frac{n}{n+2} \]
**80.** \[ \sum_{n=1}^{\infty} (1 - (-1)^n) \]
**81.** \[ \sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)} \]
**(Hint:** Use a partial fraction decomposition like that for \(\sum_{n=1}^{\infty} \frac{1}{n(n+1)}\).)
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This text includes a mention of using partial fraction decomposition for solving series convergence, a concept often explored in calculus and real analysis courses. Partial fraction decomposition is a useful algebraic tool to simplify expressions where the series terms have rational functions. This allows for easier analysis of convergence properties.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03fd5f25-eded-4046-9c65-9e89b5e28b89%2Fedbf8e99-8289-4566-92db-e50f4547fdeb%2F85ic31m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Here’s a transcription of the text as it might appear on an educational website:
---
### Series and Convergence
**78.** \( S_n = 2 - (n + 2)/2^n, n \geq 1 \)
For each of the following series, use the **sequence of partial sums** to determine whether the **series converges or diverges**.
**79.** \[ \sum_{n=1}^{\infty} \frac{n}{n+2} \]
**80.** \[ \sum_{n=1}^{\infty} (1 - (-1)^n) \]
**81.** \[ \sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)} \]
**(Hint:** Use a partial fraction decomposition like that for \(\sum_{n=1}^{\infty} \frac{1}{n(n+1)}\).)
---
This text includes a mention of using partial fraction decomposition for solving series convergence, a concept often explored in calculus and real analysis courses. Partial fraction decomposition is a useful algebraic tool to simplify expressions where the series terms have rational functions. This allows for easier analysis of convergence properties.
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