Determine whether the series is convergent or divergent. E ke-7k k = 1 O convergent divergent
Determine whether the series is convergent or divergent. E ke-7k k = 1 O convergent divergent
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Determine whether the series is convergent or divergent.**
\[ \sum_{k=1}^{\infty} ke^{-7k} \]
- ⃝ convergent
- ⃝ divergent
**Explanation:**
The given expression is an infinite series where each term is defined by the function \( ke^{-7k} \). The series requires determining whether it converges to a finite value or diverges to infinity.
**Analysis:**
To examine convergence, one may consider applying convergence tests such as the Ratio Test or Comparison Test. The exponential factor \( e^{-7k} \) implies rapid decay as \( k \) increases, which might suggest convergence. However, the factor \( k \) could affect the rate of decay. Detailed analysis and application of a suitable test are necessary to reach a conclusion about the series' behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd78c4e7c-490d-4a81-a291-db1f5fdbd52e%2F5045f1b6-a734-40e2-a82f-0a09f5056072%2F4qofg7k_processed.png&w=3840&q=75)
Transcribed Image Text:**Determine whether the series is convergent or divergent.**
\[ \sum_{k=1}^{\infty} ke^{-7k} \]
- ⃝ convergent
- ⃝ divergent
**Explanation:**
The given expression is an infinite series where each term is defined by the function \( ke^{-7k} \). The series requires determining whether it converges to a finite value or diverges to infinity.
**Analysis:**
To examine convergence, one may consider applying convergence tests such as the Ratio Test or Comparison Test. The exponential factor \( e^{-7k} \) implies rapid decay as \( k \) increases, which might suggest convergence. However, the factor \( k \) could affect the rate of decay. Detailed analysis and application of a suitable test are necessary to reach a conclusion about the series' behavior.
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