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24.1 Let fn(x) = nx 1+2 cos² nr. Prove carefully that (fn) converges √√n uniformly to 0 on R.

24.1 Let fn(x) = nx 1+2 cos² nr. Prove carefully that (fn) converges √√n uniformly to 0 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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**24.1** Let \( f_n(x) = \frac{1 + 2\cos^2 nx}{\sqrt{n}} \). Prove carefully that \( (f_n) \) converges uniformly to 0 on \( \mathbb{R} \). ### Explanation: This problem involves proving the uniform convergence of the sequence of functions \( f_n(x) \) to the zero function on the real numbers \( \mathbb{R} \). The function \( f_n(x) \) comprises a trigonometric component, \(\cos^2(nx)\), which is modulated by the sequence \(\frac{1}{\sqrt{n}}\). As \( n \) increases, the \(\frac{1}{\sqrt{n}}\) term diminishes the influence of the \( 1 + 2\cos^2(nx) \) term, leading \( f_n(x) \) to converge towards zero. Uniform convergence means that for every \(\epsilon > 0\), there exists an \( N \) such that for all \( n > N \) and all \( x \in \mathbb{R} \), the inequality \(|f_n(x)| < \epsilon\) holds.
Transcribed Image Text:**24.1** Let \( f_n(x) = \frac{1 + 2\cos^2 nx}{\sqrt{n}} \). Prove carefully that \( (f_n) \) converges uniformly to 0 on \( \mathbb{R} \). ### Explanation: This problem involves proving the uniform convergence of the sequence of functions \( f_n(x) \) to the zero function on the real numbers \( \mathbb{R} \). The function \( f_n(x) \) comprises a trigonometric component, \(\cos^2(nx)\), which is modulated by the sequence \(\frac{1}{\sqrt{n}}\). As \( n \) increases, the \(\frac{1}{\sqrt{n}}\) term diminishes the influence of the \( 1 + 2\cos^2(nx) \) term, leading \( f_n(x) \) to converge towards zero. Uniform convergence means that for every \(\epsilon > 0\), there exists an \( N \) such that for all \( n > N \) and all \( x \in \mathbb{R} \), the inequality \(|f_n(x)| < \epsilon\) holds.
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