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Determine whether the series converges. (c) (d) (e) ∞ 1 k-11+9k² k=1 2 -1 tan ¹ k 1+k² 1 Σk² sin² k k=1

Determine whether the series converges. (c) (d) (e) ∞ 1 k-11+9k² k=1 2 -1 tan ¹ k 1+k² 1 Σk² sin² k k=1

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Determine whether the series converges.** **(c)** \(\ \sum_{k=1}^{\infty} \frac{1}{1 + 9k^2} \) **(d)** \(\ \sum_{k=1}^{\infty} \frac{\tan^{-1}k}{1 + k^2} \) **(e)** \(\ \sum_{k=1}^{\infty} k^2 \sin^2 \frac{1}{k} \) For educational purposes, your task is to analyze whether each of these infinite series converges or diverges. This involves exploring the behavior of each series' terms as \( k \) approaches infinity and applying relevant convergence tests where appropriate.
Transcribed Image Text:**Determine whether the series converges.** **(c)** \(\ \sum_{k=1}^{\infty} \frac{1}{1 + 9k^2} \) **(d)** \(\ \sum_{k=1}^{\infty} \frac{\tan^{-1}k}{1 + k^2} \) **(e)** \(\ \sum_{k=1}^{\infty} k^2 \sin^2 \frac{1}{k} \) For educational purposes, your task is to analyze whether each of these infinite series converges or diverges. This involves exploring the behavior of each series' terms as \( k \) approaches infinity and applying relevant convergence tests where appropriate.
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