Determine if each series converges or diverges and state which test justifies your conclusion. When appropriate, state if the series is absolutely convergent or conditionally convergent. 00 n 2n Σ 4n? 1. 2 n+1 Σ 6. n=1 2" n=1 Σ 1 4" 2. n=1 N+\n+1 7. 3" – 2 n=1 00 3. (2n)! (-1)" 8. 2 In+5 n° n=1 n=0 00 (-1)*1 2n +1 \n+1 n 00 4. E(-1)'e -2n 9. n=0 n=1 \n+1 00 5. n° +6 3 10. 4 n=0
Determine if each series converges or diverges and state which test justifies your conclusion. When appropriate, state if the series is absolutely convergent or conditionally convergent. 00 n 2n Σ 4n? 1. 2 n+1 Σ 6. n=1 2" n=1 Σ 1 4" 2. n=1 N+\n+1 7. 3" – 2 n=1 00 3. (2n)! (-1)" 8. 2 In+5 n° n=1 n=0 00 (-1)*1 2n +1 \n+1 n 00 4. E(-1)'e -2n 9. n=0 n=1 \n+1 00 5. n° +6 3 10. 4 n=0
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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![### Convergence and Divergence of Series
Determine if each series converges or diverges and state which test justifies your conclusion. When appropriate, state if the series is absolutely convergent or conditionally convergent.
1. \( \sum_{n=1}^{\infty} \left( \frac{2n}{n+1} \right)^n \)
2. \( \sum_{n=1}^{\infty} \frac{1}{n + \sqrt{n + 1}} \)
3. \( \sum_{n=1}^{\infty} \frac{(2n)!}{n^5} \)
4. \( \sum_{n=0}^{\infty} (-1)^n e^{-2n} \)
5. \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^3 + 6} \)
6. \( \sum_{n=1}^{\infty} \frac{4n^2}{2^n} \)
7. \( \sum_{n=1}^{\infty} \frac{4^n}{3^n - 2} \)
8. \( \sum_{n=0}^{\infty} \frac{(-1)^n}{\sqrt[3]{n + 5}} \)
9. \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1} n^2}{2n^2 + 1} \)
10. \( \sum_{n=0}^{\infty} \left( \frac{\pi}{4} \right)^n \)
### Explanation of Series
- For each series, utilize appropriate convergence tests such as the Ratio Test, Root Test, Integral Test, Alternating Series Test, Direct Comparison Test, and others as applicable.
- Determine whether the series diverges or converges.
- If the series converges, specify if it is absolutely convergent or conditionally convergent.
This problem set aims to provide practice with identifying the convergence properties of infinite series using various analytical techniques, essential for calculus and real analysis courses.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf7f6c95-8471-427f-9da1-196effb3faf4%2F26026a99-267c-4f2e-9498-6e4d25452630%2F373ujih.png&w=3840&q=75)
Transcribed Image Text:### Convergence and Divergence of Series
Determine if each series converges or diverges and state which test justifies your conclusion. When appropriate, state if the series is absolutely convergent or conditionally convergent.
1. \( \sum_{n=1}^{\infty} \left( \frac{2n}{n+1} \right)^n \)
2. \( \sum_{n=1}^{\infty} \frac{1}{n + \sqrt{n + 1}} \)
3. \( \sum_{n=1}^{\infty} \frac{(2n)!}{n^5} \)
4. \( \sum_{n=0}^{\infty} (-1)^n e^{-2n} \)
5. \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^3 + 6} \)
6. \( \sum_{n=1}^{\infty} \frac{4n^2}{2^n} \)
7. \( \sum_{n=1}^{\infty} \frac{4^n}{3^n - 2} \)
8. \( \sum_{n=0}^{\infty} \frac{(-1)^n}{\sqrt[3]{n + 5}} \)
9. \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1} n^2}{2n^2 + 1} \)
10. \( \sum_{n=0}^{\infty} \left( \frac{\pi}{4} \right)^n \)
### Explanation of Series
- For each series, utilize appropriate convergence tests such as the Ratio Test, Root Test, Integral Test, Alternating Series Test, Direct Comparison Test, and others as applicable.
- Determine whether the series diverges or converges.
- If the series converges, specify if it is absolutely convergent or conditionally convergent.
This problem set aims to provide practice with identifying the convergence properties of infinite series using various analytical techniques, essential for calculus and real analysis courses.
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