Use either the divergence test or the integral test to determine whether the series converge or diverge. Explain why the series meets the hypotheses of the test you select. ∞ k=1 tan-¹k 1+k²
Use either the divergence test or the integral test to determine whether the series converge or diverge. Explain why the series meets the hypotheses of the test you select. ∞ k=1 tan-¹k 1+k²
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
![**Instructions:**
Use either the divergence test or the integral test to determine whether the series converge or diverge. Explain why the series meets the hypotheses of the test you select.
**Series:**
\[
\sum_{k=1}^{\infty} \frac{\tan^{-1} k}{1 + k^2}
\]
**Explanation:**
The series in question is the sum from \( k = 1 \) to \( \infty \) of the term \( \frac{\tan^{-1} k}{1 + k^2} \). You are asked to use either the divergence test or the integral test to analyze the convergence or divergence of this series. When using your selected test, ensure to justify clearly why the series satisfies the necessary conditions for the chosen method.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a3ba5c9-3580-45cd-ae24-18a07977dcd9%2Faa0e4bd2-4aa0-4682-9f2d-58c52a992ea9%2Fssryy_processed.png&w=3840&q=75)
Transcribed Image Text:**Instructions:**
Use either the divergence test or the integral test to determine whether the series converge or diverge. Explain why the series meets the hypotheses of the test you select.
**Series:**
\[
\sum_{k=1}^{\infty} \frac{\tan^{-1} k}{1 + k^2}
\]
**Explanation:**
The series in question is the sum from \( k = 1 \) to \( \infty \) of the term \( \frac{\tan^{-1} k}{1 + k^2} \). You are asked to use either the divergence test or the integral test to analyze the convergence or divergence of this series. When using your selected test, ensure to justify clearly why the series satisfies the necessary conditions for the chosen method.
Expert Solution
Introduction
Integral Test:
A positive term series is said to be convergent if finite
and said to be divergent if is infinite where is the positive
and monotonic decreasing sequence for all
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