Σ(-1)*+1 k=1 3k +1 3k4 + k + 6k6
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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Question
Use any method to determine whether the series converges (conditionally or absolutely). There are three possible answers: absolutely convergent, conditionally convergent or divergent.
![The expression in the image is a mathematical series described as follows:
\[
\sum_{k=1}^{\infty} (-1)^{k+1} \frac{3k^3 + 1}{3k^4 + k + 6k^6}
\]
### Explanation:
- **Summation Symbol (\(\sum\))**: This represents the infinite sum of the terms that follow, starting from \(k = 1\) and continuing indefinitely (\(\infty\)).
- **Alternating Sign \((-1)^{k+1}\)**: The term \((-1)^{k+1}\) ensures that the series alternates signs for consecutive terms. For odd \(k\), the term is positive, and for even \(k\), the term is negative.
- **Numerator (\(3k^3 + 1\))**: This is the polynomial \(3k^3 + 1\) in the numerator of the fraction.
- **Denominator (\(3k^4 + k + 6k^6\))**: This is the polynomial \(3k^4 + k + 6k^6\) in the denominator of the fraction.
This series represents an alternating, infinite series where each term is a fraction involving polynomials of \(k\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F277d8e9e-796c-4923-a9a2-4c05d5ad95df%2F6aeeeb3c-0b46-44aa-a056-b24e1272526b%2Fzixzr8o_processed.png&w=3840&q=75)
Transcribed Image Text:The expression in the image is a mathematical series described as follows:
\[
\sum_{k=1}^{\infty} (-1)^{k+1} \frac{3k^3 + 1}{3k^4 + k + 6k^6}
\]
### Explanation:
- **Summation Symbol (\(\sum\))**: This represents the infinite sum of the terms that follow, starting from \(k = 1\) and continuing indefinitely (\(\infty\)).
- **Alternating Sign \((-1)^{k+1}\)**: The term \((-1)^{k+1}\) ensures that the series alternates signs for consecutive terms. For odd \(k\), the term is positive, and for even \(k\), the term is negative.
- **Numerator (\(3k^3 + 1\))**: This is the polynomial \(3k^3 + 1\) in the numerator of the fraction.
- **Denominator (\(3k^4 + k + 6k^6\))**: This is the polynomial \(3k^4 + k + 6k^6\) in the denominator of the fraction.
This series represents an alternating, infinite series where each term is a fraction involving polynomials of \(k\).
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