Use the Comparison & Alternating Tests to determine whether or not the following series converges. If the series does converge, then estimate the sum with Sopo and find an error bound thie ostimote
Use the Comparison & Alternating Tests to determine whether or not the following series converges. If the series does converge, then estimate the sum with Sopo and find an error bound thie ostimote
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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![**Title: Understanding Series Convergence Using Comparison & Alternating Tests**
**Introduction:**
In this lesson, we will use the Comparison and Alternating Tests to determine the convergence of series. If a series is found to converge, we will estimate the sum with \( S_{1000} \) and determine an error bound for this estimate.
**Problem Statement:**
Examine the convergence of the following series:
**Equation (c):**
\[ S_{\infty} = \sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n+1}} \approx \]
**Equation (d):**
\[ S_{\infty} = \sum_{n=0}^{\infty} \frac{(-1)^n}{1 + \arctan(n)} \approx \]
**Methodology:**
- **Comparison Test:** Compare the given series to a known convergent or divergent series to determine convergence.
- **Alternating Series Test:** Check if the series satisfies the conditions for convergence of an alternating series:
1. The absolute values of the terms decrease monotonically.
2. The limit of the sequence of terms is zero as \( n \) approaches infinity.
**If the series converges:**
1. **Estimate the Sum:** Calculate the partial sum \( S_{1000} \) by summing the first 1000 terms of the series.
2. **Error Bound:** Determine an error bound for the estimation based on the nth term after \( n = 1000 \).
**Conclusion:**
Utilize these tests and methods to explore convergence and provide accurate estimates for infinite series.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7d01a9fa-2d22-41b6-ab89-d6b153d8821e%2F9c441c64-f1ad-4490-97f5-f2afffb31357%2Fckf8p7r_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Series Convergence Using Comparison & Alternating Tests**
**Introduction:**
In this lesson, we will use the Comparison and Alternating Tests to determine the convergence of series. If a series is found to converge, we will estimate the sum with \( S_{1000} \) and determine an error bound for this estimate.
**Problem Statement:**
Examine the convergence of the following series:
**Equation (c):**
\[ S_{\infty} = \sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n+1}} \approx \]
**Equation (d):**
\[ S_{\infty} = \sum_{n=0}^{\infty} \frac{(-1)^n}{1 + \arctan(n)} \approx \]
**Methodology:**
- **Comparison Test:** Compare the given series to a known convergent or divergent series to determine convergence.
- **Alternating Series Test:** Check if the series satisfies the conditions for convergence of an alternating series:
1. The absolute values of the terms decrease monotonically.
2. The limit of the sequence of terms is zero as \( n \) approaches infinity.
**If the series converges:**
1. **Estimate the Sum:** Calculate the partial sum \( S_{1000} \) by summing the first 1000 terms of the series.
2. **Error Bound:** Determine an error bound for the estimation based on the nth term after \( n = 1000 \).
**Conclusion:**
Utilize these tests and methods to explore convergence and provide accurate estimates for infinite series.
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