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So , if z< 0. et (fn)n be a sequence of functions fn : [0, 1] →R defined by fn(r) = nz Then 1. f converges to 0 pointwisely on [0, 1] since fn converges to 0 pointwisely. 2. fn converges to f = 0 uniformly on [0, 1] and thus f fn(x)dr converges to f f(x)dr. 3. fn converges to 0 pointwisely on [0, 1] but not uniformly. 4. fn doesn't converge pointwisely to 0 on [0, 1].

So , if z< 0. et (fn)n be a sequence of functions fn : [0, 1] →R defined by fn(r) = nz Then 1. f converges to 0 pointwisely on [0, 1] since fn converges to 0 pointwisely. 2. fn converges to f = 0 uniformly on [0, 1] and thus f fn(x)dr converges to f f(x)dr. 3. fn converges to 0 pointwisely on [0, 1] but not uniformly. 4. fn doesn't converge pointwisely to 0 on [0, 1].

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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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Sequence and Series

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if r S
if <a<1.
Let (fn)n be a sequence of functions fn : [0, 1]→ R defined by fn(r) =
nz
Then
1. f, converges to 0 pointwisely on [0, 1] since fn converges to 0 pointwisely.
2. fn converges tof = 0 uniformly on [0, 1] and thus f fn(x)dx converges to ſ f(x)dr.
3. fn converges to 0 pointwisely on [0, 1] but not uniformly.
4. fn doesn't converge pointwisely to 0 on [0, 1].

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