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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Question
Using the Direct Comparison Test or the Limit Comparison Test determine if the series converges or diverges.
![**Transcription for Educational Website:**
Equation 23: Evaluate the infinite series
\[
\sum_{k=1}^{\infty} \frac{\sin(1/k)}{k^2}
\]
This mathematical expression represents an infinite series where the variable \( k \) starts at 1 and increments by 1 for each term towards infinity. The function inside the series is defined by the ratio of \(\sin(1/k)\) to \(k^2\).
- **\(\sin(1/k)\)**: This part of the expression applies the sine function to the reciprocal of \( k \). As \( k \) becomes very large, \( \sin(1/k) \) approaches \(\sin(0)\), which is 0.
- **\(k^2\)**: This denotes that each term in the denominator is the square of the current value of \( k \).
The series combines these elements and sums them as \( k \) increases towards infinity. This type of series is often analyzed to determine its convergence or divergence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a36eedc-1311-4ac7-908a-8a22bc0a900c%2Ffd646b0d-24cf-401c-bbd9-2683d7a2261a%2Fkgm3xtc_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website:**
Equation 23: Evaluate the infinite series
\[
\sum_{k=1}^{\infty} \frac{\sin(1/k)}{k^2}
\]
This mathematical expression represents an infinite series where the variable \( k \) starts at 1 and increments by 1 for each term towards infinity. The function inside the series is defined by the ratio of \(\sin(1/k)\) to \(k^2\).
- **\(\sin(1/k)\)**: This part of the expression applies the sine function to the reciprocal of \( k \). As \( k \) becomes very large, \( \sin(1/k) \) approaches \(\sin(0)\), which is 0.
- **\(k^2\)**: This denotes that each term in the denominator is the square of the current value of \( k \).
The series combines these elements and sums them as \( k \) increases towards infinity. This type of series is often analyzed to determine its convergence or divergence.
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