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
How do we get the limit 1 here.
![The image displays a mathematical expression of an infinite series that needs to be analyzed for convergence. The expression is:
\[ b) \sum_{n=1}^{\infty} \frac{n(n+3)}{(n+1)(n+2)(n+5)} \]
This series is a rational expression where the numerator is \( n(n+3) \) and the denominator is the product of three consecutive integers shifted by constants, namely \( (n+1)(n+2)(n+5) \).
The task is to determine whether this series converges. To do this, one might use tests for convergence such as the Ratio Test, Root Test, or Comparison Test, examining the behavior of the series as \( n \) approaches infinity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61b75115-d70f-4fe2-af93-2076876ad69a%2Fccefa924-92af-45f0-9c6f-281c99c21400%2F1bu6f1l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image displays a mathematical expression of an infinite series that needs to be analyzed for convergence. The expression is:
\[ b) \sum_{n=1}^{\infty} \frac{n(n+3)}{(n+1)(n+2)(n+5)} \]
This series is a rational expression where the numerator is \( n(n+3) \) and the denominator is the product of three consecutive integers shifted by constants, namely \( (n+1)(n+2)(n+5) \).
The task is to determine whether this series converges. To do this, one might use tests for convergence such as the Ratio Test, Root Test, or Comparison Test, examining the behavior of the series as \( n \) approaches infinity.
![### Comparison with Divergent Series
**Task**: Compare with the divergent series
\[
\sum_{n=1}^{\infty} \frac{1}{n}
\]
**Calculation**:
The limit is calculated as follows:
\[
\rho = \lim_{{n \to \infty}} \frac{n^2(n+3)}{(n+1)(n+2)(n+5)}
\]
As \( n \to \infty \), the expression simplifies to:
\[
\rho = 1
\]
**Conclusion**: The series diverges.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61b75115-d70f-4fe2-af93-2076876ad69a%2Fccefa924-92af-45f0-9c6f-281c99c21400%2Fz6nac8u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Comparison with Divergent Series
**Task**: Compare with the divergent series
\[
\sum_{n=1}^{\infty} \frac{1}{n}
\]
**Calculation**:
The limit is calculated as follows:
\[
\rho = \lim_{{n \to \infty}} \frac{n^2(n+3)}{(n+1)(n+2)(n+5)}
\]
As \( n \to \infty \), the expression simplifies to:
\[
\rho = 1
\]
**Conclusion**: The series diverges.
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