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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determine the convergence or divergence of the series
![### Series and Summations
**a. Infinite Series**
\[ \frac{1}{5} + \frac{1}{7} + \frac{1}{9} + \frac{1}{11} + \cdots \]
This represents an infinite series where the terms are fractions with consecutive odd denominators starting from 5.
**b. Infinite Sum Notation**
\[ \sum_{n=1}^{\infty} \frac{6^n}{5^n - 1} \]
In this expression, \(\sum\) signifies summation. The terms inside the summation symbol are of the form \(\frac{6^n}{5^n - 1}\), and the summation is taken from \(n = 1\) to infinity.
**c. Infinite Sum with Polynomial Terms**
\[ \sum_{n=1}^{\infty} \frac{5n^2 + 7n + 2}{16n^5 + 9} \]
Here, the summation involves a more complex term in the form of a fraction. The numerator is a quadratic polynomial (\(5n^2 + 7n + 2\)), and the denominator is a polynomial of the fifth degree (\(16n^5 + 9\)).
**d. Complex Infinite Series**
\[ \frac{1}{1} - \frac{1 \cdot 2}{1 \cdot 4} + \frac{1 \cdot 2 \cdot 3}{1 \cdot 4 \cdot 7} - \frac{1 \cdot 2 \cdot 3 \cdot 4}{1 \cdot 4 \cdot 7 \cdot 10} + \cdots \]
This series involves products in both the numerator and the denominator. For each term, the numerator is a product of the first \(k\) natural numbers (\(1 \cdot 2 \cdot 3 \cdot \cdots \cdot k\)), and the denominator follows a pattern beginning with 1 and then increasing by 3 for each subsequent factor in the product.
### Explanation of Graphs/Diagrams (If Present)
No graphs or diagrams are provided in the image. The entire content consists of mathematical series and summations written in symbolic form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32c25cea-359b-456d-b95a-7b7f50563263%2F6d6fe6d0-a1f5-49b3-8734-c93c9a23c857%2Ftgmqfza.png&w=3840&q=75)
Transcribed Image Text:### Series and Summations
**a. Infinite Series**
\[ \frac{1}{5} + \frac{1}{7} + \frac{1}{9} + \frac{1}{11} + \cdots \]
This represents an infinite series where the terms are fractions with consecutive odd denominators starting from 5.
**b. Infinite Sum Notation**
\[ \sum_{n=1}^{\infty} \frac{6^n}{5^n - 1} \]
In this expression, \(\sum\) signifies summation. The terms inside the summation symbol are of the form \(\frac{6^n}{5^n - 1}\), and the summation is taken from \(n = 1\) to infinity.
**c. Infinite Sum with Polynomial Terms**
\[ \sum_{n=1}^{\infty} \frac{5n^2 + 7n + 2}{16n^5 + 9} \]
Here, the summation involves a more complex term in the form of a fraction. The numerator is a quadratic polynomial (\(5n^2 + 7n + 2\)), and the denominator is a polynomial of the fifth degree (\(16n^5 + 9\)).
**d. Complex Infinite Series**
\[ \frac{1}{1} - \frac{1 \cdot 2}{1 \cdot 4} + \frac{1 \cdot 2 \cdot 3}{1 \cdot 4 \cdot 7} - \frac{1 \cdot 2 \cdot 3 \cdot 4}{1 \cdot 4 \cdot 7 \cdot 10} + \cdots \]
This series involves products in both the numerator and the denominator. For each term, the numerator is a product of the first \(k\) natural numbers (\(1 \cdot 2 \cdot 3 \cdot \cdots \cdot k\)), and the denominator follows a pattern beginning with 1 and then increasing by 3 for each subsequent factor in the product.
### Explanation of Graphs/Diagrams (If Present)
No graphs or diagrams are provided in the image. The entire content consists of mathematical series and summations written in symbolic form.
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