Use the Integral Test to determine if this serics converges. Bc sure to prove that the Integral Test is valid for this series: n-1 (n2+1)²
Use the Integral Test to determine if this serics converges. Bc sure to prove that the Integral Test is valid for this series: n-1 (n2+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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![**Transcription for Educational Website:**
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**Convergence and Estimation of Series using the Integral Test**
**Problem Statement:**
1. **Integral Test for Convergence:**
Use the Integral Test to determine if the following series converges. Be sure to prove that the Integral Test is valid for this series:
\[
\sum_{n=1}^{\infty} \frac{n}{(n^2+1)^2}
\]
2. **Estimation of Series Value:**
You are going to estimate the value of
\[
\sum_{n=1}^{\infty} \frac{n}{(n^2+1)^2}
\]
by taking the first \( n \) terms: \( a_1 + a_2 + a_3 + \cdots + a_n \).
What must \( n \) be in order for your estimate to be within at most 0.01 of the exact value? (Hint: This is the same series as in question 11, so you can re-use some of the calculations.)
**Note:** The series above is the primary focus of both tasks, with the first task verifying its convergence through the Integral Test and the second task estimating its value with a specified accuracy.
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The image does not contain any graphs or diagrams. It focuses on mathematical expressions and instructions for solving a problem related to series convergence and estimation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1af4cad-3a82-431e-a6a5-54b2babd4110%2F2581b3d0-b3c6-422f-bf75-9a95fb5ac888%2F467j0zd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website:**
---
**Convergence and Estimation of Series using the Integral Test**
**Problem Statement:**
1. **Integral Test for Convergence:**
Use the Integral Test to determine if the following series converges. Be sure to prove that the Integral Test is valid for this series:
\[
\sum_{n=1}^{\infty} \frac{n}{(n^2+1)^2}
\]
2. **Estimation of Series Value:**
You are going to estimate the value of
\[
\sum_{n=1}^{\infty} \frac{n}{(n^2+1)^2}
\]
by taking the first \( n \) terms: \( a_1 + a_2 + a_3 + \cdots + a_n \).
What must \( n \) be in order for your estimate to be within at most 0.01 of the exact value? (Hint: This is the same series as in question 11, so you can re-use some of the calculations.)
**Note:** The series above is the primary focus of both tasks, with the first task verifying its convergence through the Integral Test and the second task estimating its value with a specified accuracy.
---
The image does not contain any graphs or diagrams. It focuses on mathematical expressions and instructions for solving a problem related to series convergence and estimation.
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