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3. Let fn(x) = ¹+cos(nx), x € R. Find f(x) so that fn → ƒ pointwise on R, then check whether fn → f uniformly or not on R.

3. Let fn(x) = ¹+cos(nx), x € R. Find f(x) so that fn → ƒ pointwise on R, then check whether fn → f uniformly or not on R.

Advanced Engineering Mathematics
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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**Problem 3:** Let \( f_n(x) = \frac{1 + \cos(nx)}{n} \), where \( x \in \mathbb{R} \). Find a function \( f(x) \) such that \( f_n \) converges to \( f \) pointwise on \( \mathbb{R} \). Then, check whether \( f_n \) converges to \( f \) uniformly on \( \mathbb{R} \). --- **Explanation:** The problem presents a sequence of functions \( f_n(x) \), defined as \( \frac{1 + \cos(nx)}{n} \), and asks for the pointwise limit of these functions as \( n \to \infty \). After finding the pointwise limit function \( f(x) \), the problem requires an investigation into whether this convergence is uniform over the set of real numbers \( \mathbb{R} \). --- When solving, you can start by analyzing the behavior of the cosine function as \( n \) increases and use the properties of limits. Check the uniformity of convergence by considering the \( \epsilon \)-\( N \) definition of uniform convergence.
Transcribed Image Text:**Problem 3:** Let \( f_n(x) = \frac{1 + \cos(nx)}{n} \), where \( x \in \mathbb{R} \). Find a function \( f(x) \) such that \( f_n \) converges to \( f \) pointwise on \( \mathbb{R} \). Then, check whether \( f_n \) converges to \( f \) uniformly on \( \mathbb{R} \). --- **Explanation:** The problem presents a sequence of functions \( f_n(x) \), defined as \( \frac{1 + \cos(nx)}{n} \), and asks for the pointwise limit of these functions as \( n \to \infty \). After finding the pointwise limit function \( f(x) \), the problem requires an investigation into whether this convergence is uniform over the set of real numbers \( \mathbb{R} \). --- When solving, you can start by analyzing the behavior of the cosine function as \( n \) increases and use the properties of limits. Check the uniformity of convergence by considering the \( \epsilon \)-\( N \) definition of uniform convergence.
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