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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include: 1) the name of the test used 2) the execution of the test and any conditions required for execution 3) the conditions for convergence/divergence 4) conclusion
![### Infinite Series Representation
The given mathematical expression represents an infinite series. The expression is as follows:
\[
\sum_{n=1}^{\infty} \frac{5}{\sqrt[5]{n^3}}
\]
### Explanation:
1. **Summation Notation** \(\sum\): This symbol indicates summation. It represents the addition of a sequence of terms, starting from \(n=1\) and continuing to infinity (\(\infty\)).
2. **Variable \(n\)**: This is the index of summation, starting at \(n = 1\) and increasing by 1 for each subsequent term.
3. **The Term \(\frac{5}{\sqrt[5]{n^3}}\)**:
- **Numerator (5)**: A constant value of 5.
- **Denominator**: The fifth root (indicated by \(\sqrt[5]{ }\) or \(^\frac{1}{5}\)) of \(n\) raised to the power of 3 (\(n^3\)).
### Detailed Analysis:
- **Fifth Root**: The fifth root of a number \(x\) is a number \(y\) such that \(y^5 = x\). In the given series, \(x = n^3\).
- **Denominator Simplification**:
\[
\sqrt[5]{n^3} = (n^3)^{\frac{1}{5}} = n^{\frac{3}{5}}
\]
- **Series Term**:
\[
\frac{5}{\sqrt[5]{n^3}} = \frac{5}{n^{\frac{3}{5}}}
\]
The series can thus be rewritten as:
\[
\sum_{n=1}^{\infty} \frac{5}{n^{\frac{3}{5}}}
\]
This series sums terms of the form \(\frac{5}{n^{\frac{3}{5}}}\) starting from \(n=1\) to infinity. This reveals a pattern that could be analyzed further for convergence properties or utilized in more advanced mathematical contexts such as analysis or calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97c4b753-4d37-4f67-8dde-749f0b47db81%2Fdefc7446-8195-4067-bc79-5c0d6ff63de2%2F8vughvv_processed.png&w=3840&q=75)
Transcribed Image Text:### Infinite Series Representation
The given mathematical expression represents an infinite series. The expression is as follows:
\[
\sum_{n=1}^{\infty} \frac{5}{\sqrt[5]{n^3}}
\]
### Explanation:
1. **Summation Notation** \(\sum\): This symbol indicates summation. It represents the addition of a sequence of terms, starting from \(n=1\) and continuing to infinity (\(\infty\)).
2. **Variable \(n\)**: This is the index of summation, starting at \(n = 1\) and increasing by 1 for each subsequent term.
3. **The Term \(\frac{5}{\sqrt[5]{n^3}}\)**:
- **Numerator (5)**: A constant value of 5.
- **Denominator**: The fifth root (indicated by \(\sqrt[5]{ }\) or \(^\frac{1}{5}\)) of \(n\) raised to the power of 3 (\(n^3\)).
### Detailed Analysis:
- **Fifth Root**: The fifth root of a number \(x\) is a number \(y\) such that \(y^5 = x\). In the given series, \(x = n^3\).
- **Denominator Simplification**:
\[
\sqrt[5]{n^3} = (n^3)^{\frac{1}{5}} = n^{\frac{3}{5}}
\]
- **Series Term**:
\[
\frac{5}{\sqrt[5]{n^3}} = \frac{5}{n^{\frac{3}{5}}}
\]
The series can thus be rewritten as:
\[
\sum_{n=1}^{\infty} \frac{5}{n^{\frac{3}{5}}}
\]
This series sums terms of the form \(\frac{5}{n^{\frac{3}{5}}}\) starting from \(n=1\) to infinity. This reveals a pattern that could be analyzed further for convergence properties or utilized in more advanced mathematical contexts such as analysis or calculus.
![## Series and Summation Notation
In this module, we will explore examples of infinite series and their notation.
### Example 1
\[
\sum_{n=1}^{\infty} \frac{(-1)^n 8}{3^n}
\]
This expression represents an infinite series starting from \( n = 1 \) and continuing indefinitely. Each term in the series is given by the formula:
\[
\frac{(-1)^n 8}{3^n}
\]
#### Explanation
- \(\sum\): The summation symbol indicates that we are adding up a series of terms.
- \(n=1\) to \(\infty\): Indicates that the variable \( n \) starts at 1 and increases indefinitely.
- \( \frac{(-1)^n 8}{3^n} \): The general term of the series.
### Example 2
\[
3) \sum_{n=0}^{\infty} \left(5 + (-1)^n \right)
\]
This expression represents another infinite series that starts from \( n = 0 \) and continues indefinitely. Each term in this series is given by the formula:
\[
5 + (-1)^n
\]
#### Explanation
- \(\sum\): The summation symbol indicates that we are adding up a series of terms.
- \(n=0\) to \(\infty\): Indicates that the variable \( n \) starts at 0 and increases indefinitely.
- \( 5 + (-1)^n \): The general term of the series, where the term alternates due to \((-1)^n\).
### Summary
These series are examples of how infinite summation can be represented in mathematical notation. The first example alternates in sign and diminishes as \( n \) increases, while the second example incorporates a constant term (5) with an alternating sequence.
Understanding these forms can help in recognizing patterns and properties of different types of series, particularly those involving infinite summation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97c4b753-4d37-4f67-8dde-749f0b47db81%2Fdefc7446-8195-4067-bc79-5c0d6ff63de2%2F198yq0h_processed.png&w=3840&q=75)
Transcribed Image Text:## Series and Summation Notation
In this module, we will explore examples of infinite series and their notation.
### Example 1
\[
\sum_{n=1}^{\infty} \frac{(-1)^n 8}{3^n}
\]
This expression represents an infinite series starting from \( n = 1 \) and continuing indefinitely. Each term in the series is given by the formula:
\[
\frac{(-1)^n 8}{3^n}
\]
#### Explanation
- \(\sum\): The summation symbol indicates that we are adding up a series of terms.
- \(n=1\) to \(\infty\): Indicates that the variable \( n \) starts at 1 and increases indefinitely.
- \( \frac{(-1)^n 8}{3^n} \): The general term of the series.
### Example 2
\[
3) \sum_{n=0}^{\infty} \left(5 + (-1)^n \right)
\]
This expression represents another infinite series that starts from \( n = 0 \) and continues indefinitely. Each term in this series is given by the formula:
\[
5 + (-1)^n
\]
#### Explanation
- \(\sum\): The summation symbol indicates that we are adding up a series of terms.
- \(n=0\) to \(\infty\): Indicates that the variable \( n \) starts at 0 and increases indefinitely.
- \( 5 + (-1)^n \): The general term of the series, where the term alternates due to \((-1)^n\).
### Summary
These series are examples of how infinite summation can be represented in mathematical notation. The first example alternates in sign and diminishes as \( n \) increases, while the second example incorporates a constant term (5) with an alternating sequence.
Understanding these forms can help in recognizing patterns and properties of different types of series, particularly those involving infinite summation.
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