2. Consider the following sequence of closed subintervals of [0, 1]: | 0,1][1 3][3and denote the nth8 4subinterval by I,. Now define a sequence of functions f, (x) on [0,1] byfor x in In,fn(x) =for x not in In.(a) Show that the sequence {f„(x)} converges in the mean to the zerofunction on the interval [0, 1].(b) Show that the sequence {f„(x)} does not converge pointwise at anypoint of the interval [0, 1].

Name: 2. Consider the following sequence of closed subintervals of [0, 1]:
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Description: 2. Consider the following sequence of closed subintervals of [0, 1]: | 0,1][1 3][3and denote the nth8 4subinterval by I,. Now define a sequence of functions f, (x) on [0,1] byfor x in In,fn(x) =for x not in In.(a) Show that the sequence {f„(x)} conv

The given function is fn(x)=1 for x in In,0 for x not in In. In denote the nth subinterval. The…

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2. Consider the following sequence of closed subintervals of [0, 1]: | 0, 2 1 3 3 and denote the nth 2 4 subinterval by I,. Now define a sequence of functions f, (x) on [0,1] by [1 fn(x) =- for x in I, for x not in Ip. (a) Show that the sequence {f„(x)} converges in the mean to the zero function on the interval [0, 1]. (b) Show that the sequence {f„(x)} does not converge pointwise at any point of the interval [0, 1].

2. Consider the following sequence of closed subintervals of [0, 1]: | 0, 2 1 3 3 and denote the nth 2 4 subinterval by I,. Now define a sequence of functions f, (x) on [0,1] by [1 fn(x) =- for x in I, for x not in Ip. (a) Show that the sequence {f„(x)} converges in the mean to the zero function on the interval [0, 1]. (b) Show that the sequence {f„(x)} does not converge pointwise at any point of the interval [0, 1].

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

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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2. Consider the following sequence of closed subintervals of [0, 1]: | 0,
1][1 3][3
and denote the nth
8 4
subinterval by I,. Now define a sequence of functions f, (x) on [0,1] by
for x in In,
fn(x) =
for x not in In.
(a) Show that the sequence {f„(x)} converges in the mean to the zero
function on the interval [0, 1].
(b) Show that the sequence {f„(x)} does not converge pointwise at any
point of the interval [0, 1].

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