(2k – 1)(k² – 1) (k + 1)(k² + 4)² - k=1 d.
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
Determine whether the series is convergent or divergent.
![The mathematical expression provided is a summation, which can be presented as follows:
\[
\text{d. } \sum_{k=1}^{8} \frac{(2k - 1)(k^2 - 1)}{(k + 1)(k^2 + 4)^2}
\]
This expression represents the sum of a series where each term is given by the fraction:
\[
\frac{(2k - 1)(k^2 - 1)}{(k + 1)(k^2 + 4)^2}
\]
The variable \( k \) starts at 1 and increments by 1 up to 8. Each summand is determined by inserting the current value of \( k \) into the equation.
- **Numerator:** \( (2k - 1)(k^2 - 1) \)
- The first factor \( (2k - 1) \) is a linear expression in \( k \).
- The second factor \( (k^2 - 1) \) is a quadratic expression in \( k \).
- **Denominator:** \( (k + 1)(k^2 + 4)^2 \)
- The first factor \( (k + 1) \) is a linear expression in \( k \).
- The second factor \( (k^2 + 4) \) is a quadratic expression in \( k \), raised to the power of 2.
This summation will add up all values from \( k = 1 \) to \( k = 8 \), evaluated at each step within the provided formula.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffcba434d-76d4-4036-80a1-f05d1bad2e8d%2F1c8eb108-6f25-4399-9ec9-c3e52f5186ad%2Fpcffzmp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The mathematical expression provided is a summation, which can be presented as follows:
\[
\text{d. } \sum_{k=1}^{8} \frac{(2k - 1)(k^2 - 1)}{(k + 1)(k^2 + 4)^2}
\]
This expression represents the sum of a series where each term is given by the fraction:
\[
\frac{(2k - 1)(k^2 - 1)}{(k + 1)(k^2 + 4)^2}
\]
The variable \( k \) starts at 1 and increments by 1 up to 8. Each summand is determined by inserting the current value of \( k \) into the equation.
- **Numerator:** \( (2k - 1)(k^2 - 1) \)
- The first factor \( (2k - 1) \) is a linear expression in \( k \).
- The second factor \( (k^2 - 1) \) is a quadratic expression in \( k \).
- **Denominator:** \( (k + 1)(k^2 + 4)^2 \)
- The first factor \( (k + 1) \) is a linear expression in \( k \).
- The second factor \( (k^2 + 4) \) is a quadratic expression in \( k \), raised to the power of 2.
This summation will add up all values from \( k = 1 \) to \( k = 8 \), evaluated at each step within the provided formula.
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