series Ef, to show that absolute convergence, even for all x, does not form convergence. Prove that the series na converges uniformly in every bounded interval, but does not converge for any value of x. For n=1, 2, 3, ..., x real, put fAx) =nx f.(x)- 1+ nx*" Show that (f.) converges uniformly to a function f, and that the equati
series Ef, to show that absolute convergence, even for all x, does not form convergence. Prove that the series na converges uniformly in every bounded interval, but does not converge for any value of x. For n=1, 2, 3, ..., x real, put fAx) =nx f.(x)- 1+ nx*" Show that (f.) converges uniformly to a function f, and that the equati
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:ס !ו.. lוו. 7:48
83
Rudin principal.
166 PRINCIPLES OF MATHEMATICAL ANALYSIS
5. Let
f.(x)={ sin
0.
Show that {f) converges to a continuous function, but not uniformly. Use the
series Ef. 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
fAx) =1nx
Show that {f} converges uniformly to a function f, and that the equation
f'(x) = lim fi(x)
is correct if x #0, but false if x = 0.
8. If
(x<0),
(x >0),
I(x) =
if (xn} is a sequence of distinct points of (a, b), and if Elc. converges, prove that
the series
f(x) -
,(х — х.)
(asxsb)
converges uniformly, and that f is continuous for every x + x,.
9. Let {fa) 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 such that x, x, and x € E. Is the converse of
this true?
SEQUENCES AND SERIES OF FUNCTIONS
10. Lattina (r) denote the fractional nart of the real number x (see Exercise 16. Chap. 4.
...
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