Determine how many terms are required to add to get within a 4 decimal place accuracy of the actual, for the following series (using the remainder technique for alternating series): (b) E(-1)*1- 5"
Determine how many terms are required to add to get within a 4 decimal place accuracy of the actual, for the following series (using the remainder technique for alternating series): (b) E(-1)*1- 5"
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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![### Series Approximation Problem
**Objective:**
Determine how many terms are required to achieve a 4-decimal place accuracy of the actual value for the given series, utilizing the remainder technique for alternating series.
**Given Series:**
\[
\sum_{n=3}^{\infty} (-1)^{n+1} \frac{1}{5^n}
\]
**Explanation:**
1. **Series Type:** The series provided is an alternating series, indicated by the factor \((-1)^{n+1}\).
2. **Remainder Technique for Alternating Series:**
- In alternating series, the error (or the remainder) after approximating with a finite number of terms is less than or equal to the absolute value of the first omitted term.
- To find how many terms are required for a certain accuracy (4 decimal places in this case), determine the smallest \( n \) for which the first omitted term is less than \( 0.0001 \).
3. **Series Details:**
- Each term in the series is given by \((-1)^{n+1} \times \frac{1}{5^n}\).
- The starting index \( n = 3 \) should be noted, as this affects the number of terms needed for the approximation.
The task involves using these concepts to solve for the accurate number of terms required to meet the precision criterion stated.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa40a52fa-ca63-40a5-a54d-3f60ff8c6b34%2F87b45664-5e05-4942-a1c5-b63ffd1e2023%2F4y27zmu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Series Approximation Problem
**Objective:**
Determine how many terms are required to achieve a 4-decimal place accuracy of the actual value for the given series, utilizing the remainder technique for alternating series.
**Given Series:**
\[
\sum_{n=3}^{\infty} (-1)^{n+1} \frac{1}{5^n}
\]
**Explanation:**
1. **Series Type:** The series provided is an alternating series, indicated by the factor \((-1)^{n+1}\).
2. **Remainder Technique for Alternating Series:**
- In alternating series, the error (or the remainder) after approximating with a finite number of terms is less than or equal to the absolute value of the first omitted term.
- To find how many terms are required for a certain accuracy (4 decimal places in this case), determine the smallest \( n \) for which the first omitted term is less than \( 0.0001 \).
3. **Series Details:**
- Each term in the series is given by \((-1)^{n+1} \times \frac{1}{5^n}\).
- The starting index \( n = 3 \) should be noted, as this affects the number of terms needed for the approximation.
The task involves using these concepts to solve for the accurate number of terms required to meet the precision criterion stated.
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