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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Test the series for convergence or divergence.
![The image displays a mathematical expression in the form of an infinite series. The expression is as follows:
\[
\sum_{n=1}^{\infty} (-1)^n \frac{n^2 - 1}{n^2 + 1}
\]
This series is a summation where:
- The index \(n\) starts from 1 and goes to infinity.
- The term \((-1)^n\) indicates that the series alternates in sign (positive for odd \(n\) and negative for even \(n\)).
- The fraction \(\frac{n^2 - 1}{n^2 + 1}\) is the main component of each term in the series.
This type of infinite series is commonly studied in calculus and mathematical analysis, particularly in the context of convergence and divergence of series. The behavior of the series depends on the interplay between the alternating sign and the function applied to \(n\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffcba434d-76d4-4036-80a1-f05d1bad2e8d%2Fe82e56c8-3f43-4790-8161-c54b03116a0a%2F3qykyaq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image displays a mathematical expression in the form of an infinite series. The expression is as follows:
\[
\sum_{n=1}^{\infty} (-1)^n \frac{n^2 - 1}{n^2 + 1}
\]
This series is a summation where:
- The index \(n\) starts from 1 and goes to infinity.
- The term \((-1)^n\) indicates that the series alternates in sign (positive for odd \(n\) and negative for even \(n\)).
- The fraction \(\frac{n^2 - 1}{n^2 + 1}\) is the main component of each term in the series.
This type of infinite series is commonly studied in calculus and mathematical analysis, particularly in the context of convergence and divergence of series. The behavior of the series depends on the interplay between the alternating sign and the function applied to \(n\).
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