-nx For n EN, consider the function f(x) = e on the interval [0, ∞). Let f(x) limn→∞ fn(x) for x = [0, ∞). (a) Describe the function f(x). Is continuous on [0, ∞). (b) Show that the sequence of functions (fn) does not converge uniformly on [0, ∞). (c) Does the sequence (fn) converge uniformly on the open interval (0, ∞)? (d) Show that for any a > 0, fn converges uniformly to f on the interval [a, ∞). =
-nx For n EN, consider the function f(x) = e on the interval [0, ∞). Let f(x) limn→∞ fn(x) for x = [0, ∞). (a) Describe the function f(x). Is continuous on [0, ∞). (b) Show that the sequence of functions (fn) does not converge uniformly on [0, ∞). (c) Does the sequence (fn) converge uniformly on the open interval (0, ∞)? (d) Show that for any a > 0, fn converges uniformly to f on the interval [a, ∞). =
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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Transcribed Image Text:-nx
(2) For n E N, consider the function fn(x) = = e on the interval [0, ∞). Let f(x) =
limn→∞ f (x) for x = [0, ∞).
(a) Describe the function f(x). Is it continuous on [0, ∞).
(b) Show that the sequence of functions (fn) does not converge uniformly on [0, ∞).
(c) Does the sequence (fn) converge uniformly on the open interval (0, ∞)?
(d) Show that for any a > 0, fn converges uniformly to f on the interval [a, ∞).
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