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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Find the sum of the following series
![### Infinite Series Representation
The equation shown represents an infinite series sum given by:
\[
\sum_{n=1}^{\infty} \frac{(-1)^{n-1}2^{n+2}}{5^n}
\]
Here's a breakdown of the components of the equation:
**Summation Symbol (\(\sum\))**
- The summation symbol \(\sum\) indicates that you sum over a series of terms.
- The lower limit of summation is \(n=1\) and the upper limit is \(\infty\), meaning the series starts at \(n=1\) and continues indefinitely.
**Term Components**
- The term inside the sum consists of several parts:
- \( (-1)^{n-1} \): This is an alternating factor that changes sign depending on the value of \(n\). When \(n\) is odd, this term is positive, and when \(n\) is even, it is negative.
- \(2^{n+2}\): This represents 2 raised to the power of \(n+2\).
- \(5^n\): This is 5 raised to the power of \(n\).
**Fractional Form**
- \[
\frac{(-1)^{n-1}2^{n+2}}{5^n}
\]
indicates that \((-1)^{n-1}2^{n+2}\) is divided by \(5^n\).
This expression is used typically in sequences and series topics in mathematics, particularly in calculus and analysis, to demonstrate a series that converges or diverges based on specific conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F333de6cc-3da3-4bcc-9995-6ee1769198c2%2F85d462e1-8e7c-4722-86b1-c0fe182d167c%2Fysy5o7u.jpeg&w=3840&q=75)
Transcribed Image Text:### Infinite Series Representation
The equation shown represents an infinite series sum given by:
\[
\sum_{n=1}^{\infty} \frac{(-1)^{n-1}2^{n+2}}{5^n}
\]
Here's a breakdown of the components of the equation:
**Summation Symbol (\(\sum\))**
- The summation symbol \(\sum\) indicates that you sum over a series of terms.
- The lower limit of summation is \(n=1\) and the upper limit is \(\infty\), meaning the series starts at \(n=1\) and continues indefinitely.
**Term Components**
- The term inside the sum consists of several parts:
- \( (-1)^{n-1} \): This is an alternating factor that changes sign depending on the value of \(n\). When \(n\) is odd, this term is positive, and when \(n\) is even, it is negative.
- \(2^{n+2}\): This represents 2 raised to the power of \(n+2\).
- \(5^n\): This is 5 raised to the power of \(n\).
**Fractional Form**
- \[
\frac{(-1)^{n-1}2^{n+2}}{5^n}
\]
indicates that \((-1)^{n-1}2^{n+2}\) is divided by \(5^n\).
This expression is used typically in sequences and series topics in mathematics, particularly in calculus and analysis, to demonstrate a series that converges or diverges based on specific conditions.
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