Question

Match each of the following with the correct statement. A. The series is absolutely convergent. C. The series converges, but is not absolutely convergent. D. The series diverges. 1. \( \sum_{n=1}^{\infty} \frac{(n+1)\left(4^{2}-1\right)^{n}}{4^{2 n}} \) 2. \( \sum_{n=1}^{\infty} \frac{(-1)^{n}}{2 n+2} \) 3. \( \sum_{n=1}^{\infty}(-1)^{n} \frac{\sqrt{n}}{n+3} \) 4. \( \sum_{n=1}^{\infty} \frac{(-2)^{n}}{n^{2}} \) 5. \( \sum_{n=1}^{\infty} \frac{\sin (5 n)}{n^{2}} \)

          Match each of the following with the correct statement.
A. The series is absolutely convergent.
C. The series converges, but is not absolutely convergent.
D. The series diverges.
1. \( \sum_{n=1}^{\infty} \frac{(n+1)\left(4^{2}-1\right)^{n}}{4^{2 n}} \)
2. \( \sum_{n=1}^{\infty} \frac{(-1)^{n}}{2 n+2} \)
3. \( \sum_{n=1}^{\infty}(-1)^{n} \frac{\sqrt{n}}{n+3} \)
4. \( \sum_{n=1}^{\infty} \frac{(-2)^{n}}{n^{2}} \)
5. \( \sum_{n=1}^{\infty} \frac{\sin (5 n)}{n^{2}} \)

Match each of the following with the correct statement.
A. The series is absolutely convergent.
C. The series converges, but is not absolutely convergent.
D. The series diverges.
1. ∑n=1^∞((n+1)(4^2-1)^n)/(4^2 n)
2. ∑n=1^∞((-1)^n)/(2 n+2)
3. ∑n=1^∞(-1)^n(√(n))/(n+3)
4. ∑n=1^∞((-2)^n)/(n^2)
5. ∑n=1^∞(sin (5 n))/(n^2)

Added by P V.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Andrew Noble

09/06/2023

Step 1

**Series 1: \( \sum_{n=1}^{\infty} \frac{(n+1)\left(4^{2}-1\right)^{n}}{4^{2 n}} \)** Step 2: Simplify the expression inside the series. Notice that \(4^2 - 1 = 15\), so the series becomes \( \sum_{n=1}^{\infty} \frac{(n+1)15^{n}}{16^{n}} \). Step 3: Further Show more…

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