2 3 - ( ² ) + ( 3 ) ² + ( )*²+ + converges to until you find an SN that approximates with an error less than 0.0001. 7. The series 1 + ... Calculate SN for N = 1, 2,...
2 3 - ( ² ) + ( 3 ) ² + ( )*²+ + converges to until you find an SN that approximates with an error less than 0.0001. 7. The series 1 + ... Calculate SN for N = 1, 2,...
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![### Series Convergence Problem
**Problem 7:**
The series
\[ 1 + \left(\frac{1}{5}\right) + \left(\frac{1}{5}\right)^2 + \left(\frac{1}{5}\right)^3 + \cdots \]
converges to
\[ \frac{5}{4}. \]
Calculate \( S_N \) for \( N = 1, 2, \ldots \) until you find an \( S_N \) that approximates
\[ \frac{5}{4} \]
with an error of less than 0.0001.
### Explanation:
1. **Series Definition**:
This series is a geometric series where the first term \( a = 1 \) and the common ratio \( r = \frac{1}{5} \).
2. **Convergence**:
The sum \( S \) of an infinite geometric series with \( |r| < 1 \) converges to \( \frac{a}{1 - r} \). In this case, it converges to \( \frac{1}{1 - \frac{1}{5}} = \frac{5}{4} \).
3. **Required Calculation**:
To determine \( S_N \):
- Compute the sum \( S_N \) of the first \( N \) terms of the series.
- Compare \( S_N \) with \( \frac{5}{4} \) to ensure the error is less than 0.0001.
### Example Calculation Steps:
- **First Term (N = 1)**:
\[ S_1 = 1 \]
- **Second Term (N = 2)**:
\[ S_2 = 1 + \frac{1}{5} \]
- **Third Term (N = 3)**:
\[ S_3 = 1 + \frac{1}{5} + \left(\frac{1}{5}\right)^2 \]
- **Continue**:
Continue this process until the error \(|S_N - \frac{5}{4}|\) is less than 0.0001.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37708386-d0d8-46c3-a2d8-8e337c754037%2F22677afd-8a0b-463c-9064-5c6af9c924bf%2F5okaga_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Series Convergence Problem
**Problem 7:**
The series
\[ 1 + \left(\frac{1}{5}\right) + \left(\frac{1}{5}\right)^2 + \left(\frac{1}{5}\right)^3 + \cdots \]
converges to
\[ \frac{5}{4}. \]
Calculate \( S_N \) for \( N = 1, 2, \ldots \) until you find an \( S_N \) that approximates
\[ \frac{5}{4} \]
with an error of less than 0.0001.
### Explanation:
1. **Series Definition**:
This series is a geometric series where the first term \( a = 1 \) and the common ratio \( r = \frac{1}{5} \).
2. **Convergence**:
The sum \( S \) of an infinite geometric series with \( |r| < 1 \) converges to \( \frac{a}{1 - r} \). In this case, it converges to \( \frac{1}{1 - \frac{1}{5}} = \frac{5}{4} \).
3. **Required Calculation**:
To determine \( S_N \):
- Compute the sum \( S_N \) of the first \( N \) terms of the series.
- Compare \( S_N \) with \( \frac{5}{4} \) to ensure the error is less than 0.0001.
### Example Calculation Steps:
- **First Term (N = 1)**:
\[ S_1 = 1 \]
- **Second Term (N = 2)**:
\[ S_2 = 1 + \frac{1}{5} \]
- **Third Term (N = 3)**:
\[ S_3 = 1 + \frac{1}{5} + \left(\frac{1}{5}\right)^2 \]
- **Continue**:
Continue this process until the error \(|S_N - \frac{5}{4}|\) is less than 0.0001.
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