1 on (0, 0) nl+x n=1
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
prove the uniform convergence of the following
![The image shows the mathematical expression for an infinite series, written as:
\[
\sum_{{n=1}}^{\infty} \frac{1}{n^{1+x}} \quad \text{on} \quad (0, \infty)
\]
**Description:**
- The symbol \(\sum\) denotes summation, indicating that terms will be added together in sequence.
- The limits of the summation are from \(n = 1\) to \(\infty\), meaning the series includes all natural numbers.
- The series consists of terms of the form \(\frac{1}{n^{1+x}}\), where \(x\) is a parameter that can take any value in the interval \((0, \infty)\).
- This series is a generalization of the \(p\)-series, which converges for \(1 + x > 1\).
**Series Analysis:**
- As \(x\) increases, the exponent \(1 + x\) becomes larger, which affects the convergence of the series.
- This type of series is often analyzed in the context of convergence tests, such as the \(p\)-series test or comparison test.
This type of expression is used in mathematical analysis to explore properties such as convergence and is particularly important in fields like mathematical series theory and analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae372ae2-2feb-4e9a-b164-90567cadbb97%2F0d530243-40d7-4744-825a-c17bc77aa0ac%2F6jqbrrn_processed.png&w=3840&q=75)
Transcribed Image Text:The image shows the mathematical expression for an infinite series, written as:
\[
\sum_{{n=1}}^{\infty} \frac{1}{n^{1+x}} \quad \text{on} \quad (0, \infty)
\]
**Description:**
- The symbol \(\sum\) denotes summation, indicating that terms will be added together in sequence.
- The limits of the summation are from \(n = 1\) to \(\infty\), meaning the series includes all natural numbers.
- The series consists of terms of the form \(\frac{1}{n^{1+x}}\), where \(x\) is a parameter that can take any value in the interval \((0, \infty)\).
- This series is a generalization of the \(p\)-series, which converges for \(1 + x > 1\).
**Series Analysis:**
- As \(x\) increases, the exponent \(1 + x\) becomes larger, which affects the convergence of the series.
- This type of series is often analyzed in the context of convergence tests, such as the \(p\)-series test or comparison test.
This type of expression is used in mathematical analysis to explore properties such as convergence and is particularly important in fields like mathematical series theory and analysis.
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