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Self Assessment Questions Use any method to determine whether the following series is absolutely convergent, conditionally convergent or divergent. 1. \( \sum_{n=0}^{\infty}(-1)^{n}\left(\frac{3}{n!}\right) \) 4. \( \sum_{n=3}^{\infty}(-1)^{n+1} \frac{3^{n}}{n!} \) 2. \( \sum_{n=0}^{\infty}(-1)^{n} 2^{n} \) 5. \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{2 n+1} \) 3. \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{n^{2}+1} \) 6. \( \sum_{n=6}^{\infty}(-1)^{n} \frac{n 2^{n}}{3^{n}} \)

          Self Assessment Questions
Use any method to determine whether the following series is absolutely convergent, conditionally convergent or divergent.
1. \( \sum_{n=0}^{\infty}(-1)^{n}\left(\frac{3}{n!}\right) \)
4. \( \sum_{n=3}^{\infty}(-1)^{n+1} \frac{3^{n}}{n!} \)
2. \( \sum_{n=0}^{\infty}(-1)^{n} 2^{n} \)
5. \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{2 n+1} \)
3. \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{n^{2}+1} \)
6. \( \sum_{n=6}^{\infty}(-1)^{n} \frac{n 2^{n}}{3^{n}} \)

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Self Assessment Questions
Use any method to determine whether the following series is absolutely convergent, conditionally convergent or divergent.
1. ∑n=0^∞(-1)^n((3)/(n!))
4. ∑n=3^∞(-1)^n+1(3^n)/(n!)
2. ∑n=0^∞(-1)^n 2^n
5. ∑n=1^∞(-1)^n+1(n)/(2 n+1)
3. ∑n=1^∞(-1)^n+1(n)/(n^2+1)
6. ∑n=6^∞(-1)^n(n 2^n)/(3^n)

Added by Jessica A.

Calculus

Earl W. Swokowski 5th Edition

Chapter 11

Instant Answer

Solved by Expert Mukesh Devi

Step 1

- This is an alternating series. To determine convergence, use the Alternating Series Test. - Check if \( \frac{3}{n!} \) is decreasing and approaches 0 as \( n \to \infty \). - Since \( n! \) grows very fast, \( \frac{3}{n!} \to 0 \) as \( n \to \infty \). - The Show more…

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Self Assessment Questions Use any method to determine whether the following series is absolutely convergent, conditionally convergent or divergent. 1. \( \sum_{n=0}^{\infty}(-1)^{n}\left(\frac{3}{n!}\right) \) 4. \( \sum_{n=3}^{\infty}(-1)^{n+1} \frac{3^{n}}{n!} \) 2. \( \sum_{n=0}^{\infty}(-1)^{n} 2^{n} \) 5. \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{2 n+1} \) 3. \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{n^{2}+1} \) 6. \( \sum_{n=6}^{\infty}(-1)^{n} \frac{n 2^{n}}{3^{n}} \)

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