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. ### 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.

Answered: Image /qna-images/answer/ccefa924-92af-45f0-9c6f-281c99c21400.jpg

Answered: n (n+3) b) Compare with the divergent…

n (n+3) b) Compare with the divergent series = lim- (n+1)(n+2)(n+5) = 1; diverges

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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