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
Test for convergence/divergence, State the theorems used.
![### Series and Their Convergence
Consider the following infinite series:
#### Problem r)
\[ \sum_{n=1}^{\infty} \frac{(-1)^n (2n+1)}{n^2 + 3} \]
This series involves an alternating term \((-1)^n\), which means the signs of the terms alternate as \(n\) increases. The terms consist of fractions where the numerator is \((2n + 1)\), and the denominator is \((n^2 + 3)\).
#### Problem s)
\[ \sum_{n=1}^{\infty} \frac{(-1)^n \sqrt{2n + 5}}{n^3 + 2n + 4} \]
This series also has an alternating term \((-1)^n\). The numerator is \(\sqrt{(2n + 5)}\), a square root function, and the denominator is a combination of cubic and linear terms \((n^3 + 2n + 4)\).
#### Problem t)
\[ \sum_{n=1}^{\infty} \frac{\sin(n)}{n^3} \]
In this series, the numerator is the sine function \(\sin(n)\), and the denominator is \(n^3\), a cubic function.
### Explanation Diagrams and Graphs
There are no associated graphs or diagrams included with the text. If visual aid is needed, it might involve plotting the terms of each series to observe their behavior as \(n\) increases, their convergence points, or the nature of their alternating signs. In particular:
- A plot of \(\frac{(-1)^n (2n+1)}{n^2 + 3}\) for several values of \(n\) would show the alternating series and how the terms decrease in magnitude.
- For \(\frac{(-1)^n \sqrt{2n + 5}}{n^3 + 2n + 4}\), you could observe the effect of the square root on the numerator and the rapid growth of the cubic term in the denominator.
- The series \(\frac{\sin(n)}{n^3}\) might show the periodic behavior of the sine function divided by a rapidly increasing cubic term, likely illustrating the terms getting smaller in magnitude.
These series are typically analyzed for convergence using tests such as the Alternating Series Test](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc787ae5f-ecbe-4215-b4d0-f569392eb050%2F1ce5eb0a-3e93-4add-893d-0fb5cf87c45d%2F4okp5qe_processed.png&w=3840&q=75)
Transcribed Image Text:### Series and Their Convergence
Consider the following infinite series:
#### Problem r)
\[ \sum_{n=1}^{\infty} \frac{(-1)^n (2n+1)}{n^2 + 3} \]
This series involves an alternating term \((-1)^n\), which means the signs of the terms alternate as \(n\) increases. The terms consist of fractions where the numerator is \((2n + 1)\), and the denominator is \((n^2 + 3)\).
#### Problem s)
\[ \sum_{n=1}^{\infty} \frac{(-1)^n \sqrt{2n + 5}}{n^3 + 2n + 4} \]
This series also has an alternating term \((-1)^n\). The numerator is \(\sqrt{(2n + 5)}\), a square root function, and the denominator is a combination of cubic and linear terms \((n^3 + 2n + 4)\).
#### Problem t)
\[ \sum_{n=1}^{\infty} \frac{\sin(n)}{n^3} \]
In this series, the numerator is the sine function \(\sin(n)\), and the denominator is \(n^3\), a cubic function.
### Explanation Diagrams and Graphs
There are no associated graphs or diagrams included with the text. If visual aid is needed, it might involve plotting the terms of each series to observe their behavior as \(n\) increases, their convergence points, or the nature of their alternating signs. In particular:
- A plot of \(\frac{(-1)^n (2n+1)}{n^2 + 3}\) for several values of \(n\) would show the alternating series and how the terms decrease in magnitude.
- For \(\frac{(-1)^n \sqrt{2n + 5}}{n^3 + 2n + 4}\), you could observe the effect of the square root on the numerator and the rapid growth of the cubic term in the denominator.
- The series \(\frac{\sin(n)}{n^3}\) might show the periodic behavior of the sine function divided by a rapidly increasing cubic term, likely illustrating the terms getting smaller in magnitude.
These series are typically analyzed for convergence using tests such as the Alternating Series Test
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