Determine whether the following series converges. Justify your answer. 1 Σ 6k k=1 √√6ke
Determine whether the following series converges. Justify your answer. 1 Σ 6k k=1 √√6ke
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 following series converges. Justify your answer.**
\[
\sum_{k=1}^{\infty} \frac{1}{\sqrt{6k} \cdot e^{\sqrt{6k}}}
\]
### Explanation:
This series is an infinite sum where each term is given by the expression \(\frac{1}{\sqrt{6k} \cdot e^{\sqrt{6k}}}\).
To determine whether the series converges, consider using tests such as the Comparison Test or the Ratio Test with a similar known convergent or divergent series. The exponential part in the denominator suggests rapid decay as \(k\) increases, which may imply convergence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6ed396a-58be-4ac0-9d00-206a35697371%2F4fffd17d-b14c-46ea-a565-20883f9024b3%2Fa2odf8h_processed.png&w=3840&q=75)
Transcribed Image Text:**Determine whether the following series converges. Justify your answer.**
\[
\sum_{k=1}^{\infty} \frac{1}{\sqrt{6k} \cdot e^{\sqrt{6k}}}
\]
### Explanation:
This series is an infinite sum where each term is given by the expression \(\frac{1}{\sqrt{6k} \cdot e^{\sqrt{6k}}}\).
To determine whether the series converges, consider using tests such as the Comparison Test or the Ratio Test with a similar known convergent or divergent series. The exponential part in the denominator suggests rapid decay as \(k\) increases, which may imply convergence.
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