Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![### Problem Statement from the Image:
**Write the first five terms of an infinite geometric series that converges to 2.**
### Explanation for Educational Website:
This is a mathematical problem that requires identifying the first five terms of an infinite geometric series, which is a sequence where each term is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The series must also converge to a limit of 2.
**Key Concepts:**
1. **Infinite Geometric Series:**
- A series of the form \( a + ar + ar^2 + ar^3 + \cdots \), where \( a \) is the first term and \( r \) is the common ratio.
2. **Convergence of Geometric Series:**
- An infinite geometric series \( a + ar + ar^2 + ar^3 + \cdots \) converges if the absolute value of the common ratio \( |r| < 1 \).
- The sum \( S \) of an infinite geometric series is given by \( S = \frac{a}{1 - r} \) where \( |r| < 1 \).
### Solving the Problem:
Given that the series converges to 2, we use the formula for the sum of an infinite geometric series to set up the equation:
\[ S = \frac{a}{1 - r} = 2 \]
Let's find an example:
1. Choose the first term \( a \) and a common ratio \( r \) such that \( |r| < 1 \).
2. Solve for the values of \( a \) and \( r \) under the condition that \( \frac{a}{1 - r} = 2 \).
One potential solution is:
- Let \( a = 1 \) and \( r = \frac{1}{2} \).
Then the series is:
\[ 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots \]
**First five terms of the series:**
\[ 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16} \]
This series converges to 2 because:
\[ \frac{1}{1 - \frac{1}{2](https://content.bartleby.com/qna-images/question/5c941f29-ddef-4885-9d8b-99e8246804cb/5c548878-bcbe-4eec-b018-1ebd624315c3/zhwkxf.jpeg)
Transcribed Image Text:### Problem Statement from the Image:
**Write the first five terms of an infinite geometric series that converges to 2.**
### Explanation for Educational Website:
This is a mathematical problem that requires identifying the first five terms of an infinite geometric series, which is a sequence where each term is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The series must also converge to a limit of 2.
**Key Concepts:**
1. **Infinite Geometric Series:**
- A series of the form \( a + ar + ar^2 + ar^3 + \cdots \), where \( a \) is the first term and \( r \) is the common ratio.
2. **Convergence of Geometric Series:**
- An infinite geometric series \( a + ar + ar^2 + ar^3 + \cdots \) converges if the absolute value of the common ratio \( |r| < 1 \).
- The sum \( S \) of an infinite geometric series is given by \( S = \frac{a}{1 - r} \) where \( |r| < 1 \).
### Solving the Problem:
Given that the series converges to 2, we use the formula for the sum of an infinite geometric series to set up the equation:
\[ S = \frac{a}{1 - r} = 2 \]
Let's find an example:
1. Choose the first term \( a \) and a common ratio \( r \) such that \( |r| < 1 \).
2. Solve for the values of \( a \) and \( r \) under the condition that \( \frac{a}{1 - r} = 2 \).
One potential solution is:
- Let \( a = 1 \) and \( r = \frac{1}{2} \).
Then the series is:
\[ 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots \]
**First five terms of the series:**
\[ 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16} \]
This series converges to 2 because:
\[ \frac{1}{1 - \frac{1}{2
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