3. Let g (0, ∞) → R and for every n ≥ 1, fn: (0, ∞) → R. For every 0 < t < T < ∞, assume that g € Roc[t, T] and for every n ≥ 1, fn € Rioc[t, T]. If for every n ≥ 1, |fn| ≤ 9, fnf uniformly on every compact subset of (0,00) and g = R(0, 0), prove that lim Soto Jm.² ² Sn = √ ² ² 1 f.
3. Let g (0, ∞) → R and for every n ≥ 1, fn: (0, ∞) → R. For every 0 < t < T < ∞, assume that g € Roc[t, T] and for every n ≥ 1, fn € Rioc[t, T]. If for every n ≥ 1, |fn| ≤ 9, fnf uniformly on every compact subset of (0,00) and g = R(0, 0), prove that lim Soto Jm.² ² Sn = √ ² ² 1 f.
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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![3. Let \( g : (0, \infty) \to \mathbb{R} \) and for every \( n \geq 1 \), \( f_n : (0, \infty) \to \mathbb{R} \). For every \( 0 < t < T < \infty \), assume that \( g \in \mathcal{R}_{\text{loc}}[t, T] \) and for every \( n \geq 1 \), \( f_n \in \mathcal{R}_{\text{loc}}[t, T] \). If for every \( n \geq 1 \), \( |f_n| \leq g \), \( f_n \to f \) uniformly on every compact subset of \( (0, \infty) \) and \( g \in \mathcal{R}(0, \infty) \), prove that
\[
\lim_{n \to \infty} \int_{0}^{\infty} f_n = \int_{0}^{\infty} f.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabba7d05-e030-4d49-ac3b-b588659cd1ab%2F6b5b821e-3113-4486-bf8b-7094eb02f113%2F6i98sau_processed.png&w=3840&q=75)
Transcribed Image Text:3. Let \( g : (0, \infty) \to \mathbb{R} \) and for every \( n \geq 1 \), \( f_n : (0, \infty) \to \mathbb{R} \). For every \( 0 < t < T < \infty \), assume that \( g \in \mathcal{R}_{\text{loc}}[t, T] \) and for every \( n \geq 1 \), \( f_n \in \mathcal{R}_{\text{loc}}[t, T] \). If for every \( n \geq 1 \), \( |f_n| \leq g \), \( f_n \to f \) uniformly on every compact subset of \( (0, \infty) \) and \( g \in \mathcal{R}(0, \infty) \), prove that
\[
\lim_{n \to \infty} \int_{0}^{\infty} f_n = \int_{0}^{\infty} f.
\]
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