k=3 k+1 (k - 2)³
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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Use any convergence test to determine whether the series converges or diverges. Explain how the series meets the hypotheses of the test you select.
![The image shows a mathematical expression representing an infinite series:
\[
\sum_{k=3}^{\infty} \frac{k+1}{(k-2)^3}
\]
This series begins with \( k = 3 \) and extends to infinity. The general term of the series is given by the expression:
\[
\frac{k+1}{(k-2)^3}
\]
In this expression:
- \( k \) is the variable of summation.
- The numerator \( k+1 \) increases linearly as \( k \) increases.
- The denominator \( (k-2)^3 \) is a cubic expression, which grows rapidly as \( k \) increases.
The series explores the sum of these terms from \( k = 3 \) to infinity. This type of series can be analyzed for convergence or divergence using various mathematical tests.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a3ba5c9-3580-45cd-ae24-18a07977dcd9%2F7247a70d-afa6-4800-9c8f-f48e4556eef9%2Fglzqxc9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image shows a mathematical expression representing an infinite series:
\[
\sum_{k=3}^{\infty} \frac{k+1}{(k-2)^3}
\]
This series begins with \( k = 3 \) and extends to infinity. The general term of the series is given by the expression:
\[
\frac{k+1}{(k-2)^3}
\]
In this expression:
- \( k \) is the variable of summation.
- The numerator \( k+1 \) increases linearly as \( k \) increases.
- The denominator \( (k-2)^3 \) is a cubic expression, which grows rapidly as \( k \) increases.
The series explores the sum of these terms from \( k = 3 \) to infinity. This type of series can be analyzed for convergence or divergence using various mathematical tests.
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