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. n 2n 4n2 00 Σ 1. n+1 2" n=1 n=1 1 00 4" 7. n=1 N+ Vn+1 3" – 2 n=1 ¿ (2n)! (-1) 8. 00 n° n=1 In+5 (-1)*"n² n+1 4. 9. n=1 2n² +1 n=0 00 n+1 IT 5. n° +6 10. n=1 4 n=0 6. 2. 3.
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. n 2n 4n2 00 Σ 1. n+1 2" n=1 n=1 1 00 4" 7. n=1 N+ Vn+1 3" – 2 n=1 ¿ (2n)! (-1) 8. 00 n° n=1 In+5 (-1)*"n² n+1 4. 9. n=1 2n² +1 n=0 00 n+1 IT 5. n° +6 10. n=1 4 n=0 6. 2. 3.
Advanced Engineering Mathematics
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![### Series Convergence Determination
#### Problem Statement:
For each of the following series, determine whether it converges or diverges and state which test justifies your conclusion. When appropriate, indicate 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 Tests:
- **Comparison Test**: Compare the given series to another series whose convergence is known.
- **Ratio Test**: Uses the limit of the ratio of successive terms.
- **Root Test**: Uses the limit of the nth root of the nth term.
- **Alternating Series Test (Leibniz's test)**: For series in the form \(\sum (-1)^n b_n\) where \(b_n\) is positive, decreasing, and tends to zero.
- **Integral Test**: Involves finding the integral of the continuous comparable function.
- **p-Series Test**: Series of the form \(\sum \frac{1}{n^p}\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf7f6c95-8471-427f-9da1-196effb3faf4%2F8c8ebe2c-8789-4689-b4b4-962a411548f7%2Fypvkzn.png&w=3840&q=75)
Transcribed Image Text:### Series Convergence Determination
#### Problem Statement:
For each of the following series, determine whether it converges or diverges and state which test justifies your conclusion. When appropriate, indicate 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 Tests:
- **Comparison Test**: Compare the given series to another series whose convergence is known.
- **Ratio Test**: Uses the limit of the ratio of successive terms.
- **Root Test**: Uses the limit of the nth root of the nth term.
- **Alternating Series Test (Leibniz's test)**: For series in the form \(\sum (-1)^n b_n\) where \(b_n\) is positive, decreasing, and tends to zero.
- **Integral Test**: Involves finding the integral of the continuous comparable function.
- **p-Series Test**: Series of the form \(\sum \frac{1}{n^p}\)
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