(ii) Find the pointwise limit of the sequence. (iii) Show that the convergence is uniform on any bounded subset of R.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Number 1...(ii)..(iii)...is real analysis
EXERCISE 01
of 1
SMTA031
1. Consider a sequence of functions defined on R by
r |_nr
f(x)=
FEBRUARY 2023
(i) Sketch the graphs of fi. 2. f3 and f₁ on the same set of axes.
(ii) Find the pointwise limit of the sequence.
(iii) Show that the convergence is uniform on any bounded subset of R.
2. Show that the series of functions defined by
Σfn(x) = (1 − x) + x(1 − x) + x²(1 − x) + ...
converges pointwise on [0. 1]. Argue that the convergence is not uniform.
3. Find the pointwise limit of the sequence of functions defined on R by the following
and determine whether or not the convergence is uniform.
111 LE TC 0. T
Transcribed Image Text:EXERCISE 01 of 1 SMTA031 1. Consider a sequence of functions defined on R by r |_nr f(x)= FEBRUARY 2023 (i) Sketch the graphs of fi. 2. f3 and f₁ on the same set of axes. (ii) Find the pointwise limit of the sequence. (iii) Show that the convergence is uniform on any bounded subset of R. 2. Show that the series of functions defined by Σfn(x) = (1 − x) + x(1 − x) + x²(1 − x) + ... converges pointwise on [0. 1]. Argue that the convergence is not uniform. 3. Find the pointwise limit of the sequence of functions defined on R by the following and determine whether or not the convergence is uniform. 111 LE TC 0. T
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