Q7 Real analysis by Walton rudin 1:34 ill O28Rudin principal.166 PRINCIPLES OF MATHEMATICAL ANALYSIS5. Let10f(x)={ sin"Show that {f} converges to a continuous function, but not uniformly. Use theseries E f. to show that absolute convergence, even for all x, does not imply uni-form convergence.6. Prove that the seriesconverges uniformly in every bounded interval, but does not converge absolutelyfor any value of x.7. For n= 1, 2, 3, ..., x real, put1+ nx²Show that {f} converges uniformly to a function f, and that the equationf'(x) = lim f(x)is correct if x *0, but false if x =0.8. Ifjo(x<0),I(x) =1(x>0),if (xn} is a sequence of distinct points of (a, b), and if Elc. converges, prove thatthe seriesS(x) =Ec. I(x – x.)(asxsb)converges uniformly, and that f is continuous for every x + x..9. Let {f.) be a sequence of continuous functions which converges uniformly to afunction f on a set E. Prove thatlim f.(x,) = f(x)for every sequence of points x, E E such that x →x, and x e E. Is the converse ofthis true?SEQUENCES AND SERIES OF FUNCTIONS10. Letting (x) denote the fractional part of the real number x (see Exercise 16, Ct.for the definition), consider the function(nx)S(x) =(x real).

Question 7) Given, fn(x)=x1+nx2 We have to show that fn(x) converges uniformly to a function f. We…

Answered: 7. For n-1, 2, 3, ..., x real, put SAx)…

7. For n-1, 2, 3, ..., x real, put SAx) = 1+ nx² ' Show that (S) converges uniformly to a function f, and that the equation S'(x) = limf(x) %3D

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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Question

Q7 Real analysis by Walton rudin

1:34 ill O
28
Rudin principal.
166 PRINCIPLES OF MATHEMATICAL ANALYSIS
5. Let
10
f(x)={ sin"
Show that {f} converges to a continuous function, but not uniformly. Use the
series E f. to show that absolute convergence, even for all x, does not imply uni-
form convergence.
6. Prove that the series
converges uniformly in every bounded interval, but does not converge absolutely
for any value of x.
7. For n= 1, 2, 3, ..., x real, put
1+ nx²
Show that {f} converges uniformly to a function f, and that the equation
f'(x) = lim f(x)
is correct if x *0, but false if x =0.
8. If
jo
(x<0),
I(x) =
1
(x>0),
if (xn} is a sequence of distinct points of (a, b), and if Elc. converges, prove that
the series
S(x) =
Ec. I(x – x.)
(asxsb)
converges uniformly, and that f is continuous for every x + x..
9. Let {f.) be a sequence of continuous functions which converges uniformly to a
function f on a set E. Prove that
lim f.(x,) = f(x)
for every sequence of points x, E E such that x →x, and x e E. Is the converse of
this true?
SEQUENCES AND SERIES OF FUNCTIONS
10. Letting (x) denote the fractional part of the real number x (see Exercise 16, Ct.
for the definition), consider the function
(nx)
S(x) =
(x real).

Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.

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