Determine whether the following series converges. Justify your answer. (-1)kk Σ 6 k=1 11k +1 M8 O ... O A. The series is a geometric series with common ratio so the series converges B. The terms of the series are alternating and the limit of their absolute values is Series Test. OC. The series is a p-series with p = D. The limit of the terms of the series is by the properties of a geometric series. so the series diverges by the Alternating so the series converges by the properties of a p-series. so the series diverges by the Divergence Test.
Determine whether the following series converges. Justify your answer. (-1)kk Σ 6 k=1 11k +1 M8 O ... O A. The series is a geometric series with common ratio so the series converges B. The terms of the series are alternating and the limit of their absolute values is Series Test. OC. The series is a p-series with p = D. The limit of the terms of the series is by the properties of a geometric series. so the series diverges by the Alternating so the series converges by the properties of a p-series. so the series diverges by the Divergence Test.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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The second photo is the rest of multiple choice
![**Question:**
Determine the convergence or divergence of the series using the appropriate test.
**Options:**
- **D.** The limit of the terms of the series is [blank], so the series diverges by the Divergence Test.
- **E.** The series is a geometric series with common ratio [blank], so the series diverges by the properties of a geometric series.
- **F.** The terms of the series alternate in sign, are nonincreasing in magnitude, and the limit of their absolute values is [blank], so the series converges by the Alternating Series Test.
**Note:**
The screenshot displays part of an educational exercise related to series convergence tests in calculus. The options suggest different methods (Divergence Test, properties of geometric series, Alternating Series Test) for determining if a series converges or diverges. The blanks indicate spaces where specific values or conditions would be filled in during the full solution of a problem.
Beneath these options, there are buttons labeled "Example," "Textbook," "Clear all," and "Check answer," indicating interactive elements likely intended for user engagement on an educational platform. The bottom part of the image shows the dock of a MacBook with various application icons.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcef3f1b2-56cd-4f45-949e-fc7a38ef994c%2Feb529719-b627-4a5a-b1dd-e78e858ca7ad%2F9ure65s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
Determine the convergence or divergence of the series using the appropriate test.
**Options:**
- **D.** The limit of the terms of the series is [blank], so the series diverges by the Divergence Test.
- **E.** The series is a geometric series with common ratio [blank], so the series diverges by the properties of a geometric series.
- **F.** The terms of the series alternate in sign, are nonincreasing in magnitude, and the limit of their absolute values is [blank], so the series converges by the Alternating Series Test.
**Note:**
The screenshot displays part of an educational exercise related to series convergence tests in calculus. The options suggest different methods (Divergence Test, properties of geometric series, Alternating Series Test) for determining if a series converges or diverges. The blanks indicate spaces where specific values or conditions would be filled in during the full solution of a problem.
Beneath these options, there are buttons labeled "Example," "Textbook," "Clear all," and "Check answer," indicating interactive elements likely intended for user engagement on an educational platform. The bottom part of the image shows the dock of a MacBook with various application icons.
![**Determine whether the following series converges. Justify your answer.**
\[
\sum_{k=1}^{\infty} \frac{(-1)^k \cdot k}{11^k \cdot 6^{k+1}}
\]
**Options:**
- **A.** The series is a geometric series with common ratio \(\square\), so the series converges by the properties of a geometric series.
- **B.** The terms of the series are alternating and the limit of their absolute values is \(\square\), so the series diverges by the Alternating Series Test.
- **C.** The series is a p-series with \(p = \square\), so the series converges by the properties of a p-series.
- **D.** The limit of the terms of the series is \(\square\), so the series diverges by the Divergence Test.
There are no graphs or diagrams in the image.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcef3f1b2-56cd-4f45-949e-fc7a38ef994c%2Feb529719-b627-4a5a-b1dd-e78e858ca7ad%2Fy59w1z9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determine whether the following series converges. Justify your answer.**
\[
\sum_{k=1}^{\infty} \frac{(-1)^k \cdot k}{11^k \cdot 6^{k+1}}
\]
**Options:**
- **A.** The series is a geometric series with common ratio \(\square\), so the series converges by the properties of a geometric series.
- **B.** The terms of the series are alternating and the limit of their absolute values is \(\square\), so the series diverges by the Alternating Series Test.
- **C.** The series is a p-series with \(p = \square\), so the series converges by the properties of a p-series.
- **D.** The limit of the terms of the series is \(\square\), so the series diverges by the Divergence Test.
There are no graphs or diagrams in the image.
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