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Suppose for nN, fn(x) = ne-* sin (2) x where x € (0, ∞). (a) (. ) Find a function g: (0, ∞) → R such that fn(x)| ≤ g(x) for all x € (0, ∞) and n € N, and g is Lebesgue integrable on (0, ∞). (b 5) Using (a), prove that fn is Lebesgue integrable on (0, ∞) for all n N and lim n→∞ (0,00) fn du = lim n-x S fn(x) dx = 1 (Note: fn(x) dx is an improper Riemann integral. You need to justify that this improper Riemann integral equals the corresponding Lebesgue integral on (0, ∞).)

Suppose for nN, fn(x) = ne-* sin (2) x where x € (0, ∞). (a) (. ) Find a function g: (0, ∞) → R such that fn(x)| ≤ g(x) for all x € (0, ∞) and n € N, and g is Lebesgue integrable on (0, ∞). (b 5) Using (a), prove that fn is Lebesgue integrable on (0, ∞) for all n N and lim n→∞ (0,00) fn du = lim n-x S fn(x) dx = 1 (Note: fn(x) dx is an improper Riemann integral. You need to justify that this improper Riemann integral equals the corresponding Lebesgue integral on (0, ∞).)

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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ne-* sin (2) x Suppose for n N, fn(x) = where x € (0, ∞). (a) (. ) Find a function g: (0, ∞) → R such that fn(x)| ≤ g(x) for all x € (0, ∞) and n € N, and g is Lebesgue integrable on (0, ∞). (b 5) Using (a), prove that fn is Lebesgue integrable on (0, ∞) for all n N and lim n→∞ (0,00) fn du = lim n-x S fn(x) dx = 1 (Note: fn(x) dx is an improper Riemann integral. You need to justify that this improper Riemann integral equals the corresponding Lebesgue integral on (0, ∞).)
Transcribed Image Text:ne-* sin (2) x Suppose for n N, fn(x) = where x € (0, ∞). (a) (. ) Find a function g: (0, ∞) → R such that fn(x)| ≤ g(x) for all x € (0, ∞) and n € N, and g is Lebesgue integrable on (0, ∞). (b 5) Using (a), prove that fn is Lebesgue integrable on (0, ∞) for all n N and lim n→∞ (0,00) fn du = lim n-x S fn(x) dx = 1 (Note: fn(x) dx is an improper Riemann integral. You need to justify that this improper Riemann integral equals the corresponding Lebesgue integral on (0, ∞).)
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