Determine whether the series 00 n=0 1 OD. n Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The series diverges because lim converges or diverges. If it converges, find its sum. 1 n∞ √7 n #0 or fails to exist. OB. The series converges because it is a geometric series with |r|<1. The sum of the series is (Type an exact answer, using radicals as needed.) OC. The series diverges because it is a geometric series with |r| 21. n = 0. The sum of the series is. 1 The series converges because lim (+) n→∞o (Type an exact answer, using radicals as needed.)
Determine whether the series 00 n=0 1 OD. n Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The series diverges because lim converges or diverges. If it converges, find its sum. 1 n∞ √7 n #0 or fails to exist. OB. The series converges because it is a geometric series with |r|<1. The sum of the series is (Type an exact answer, using radicals as needed.) OC. The series diverges because it is a geometric series with |r| 21. n = 0. The sum of the series is. 1 The series converges because lim (+) n→∞o (Type an exact answer, using radicals as needed.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Determine whether the series**
\[
\sum_{n=0}^{\infty} \left( \frac{1}{\sqrt{7}} \right)^n
\]
**converges or diverges. If it converges, find its sum.**
---
**Select the correct choice below and, if necessary, fill in the answer box within your choice.**
- **A.** The series diverges because \(\lim_{n \to \infty} \left( \frac{1}{\sqrt{7}} \right)^n \neq 0\) or fails to exist.
- **B.** The series converges because it is a geometric series with \(|r| < 1\). The sum of the series is \([ \text{Type an exact answer, using radicals as needed.} ]\).
- **C.** The series diverges because it is a geometric series with \(|r| \geq 1\).
- **D.** The series converges because \(\lim_{n \to \infty} \left( \frac{1}{\sqrt{7}} \right)^n = 0\). The sum of the series is \([ \text{Type an exact answer, using radicals as needed.} ]\).
---
**Note**: The expression involves analyzing a geometric series and determining convergence based on the common ratio \(r\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87314332-e2f5-4784-8bd6-50bd771df639%2F19e36724-920b-41c4-a74c-cd4d8a47add2%2Fliuy4b8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determine whether the series**
\[
\sum_{n=0}^{\infty} \left( \frac{1}{\sqrt{7}} \right)^n
\]
**converges or diverges. If it converges, find its sum.**
---
**Select the correct choice below and, if necessary, fill in the answer box within your choice.**
- **A.** The series diverges because \(\lim_{n \to \infty} \left( \frac{1}{\sqrt{7}} \right)^n \neq 0\) or fails to exist.
- **B.** The series converges because it is a geometric series with \(|r| < 1\). The sum of the series is \([ \text{Type an exact answer, using radicals as needed.} ]\).
- **C.** The series diverges because it is a geometric series with \(|r| \geq 1\).
- **D.** The series converges because \(\lim_{n \to \infty} \left( \frac{1}{\sqrt{7}} \right)^n = 0\). The sum of the series is \([ \text{Type an exact answer, using radicals as needed.} ]\).
---
**Note**: The expression involves analyzing a geometric series and determining convergence based on the common ratio \(r\).
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