Q5 "Real analysis "principal of Mathematical Analysis by rudin uniformly on E. If, in addition, {fa) and (g.) are sequences of bounded functions,prove that {fag,} converges uniformly on E.3. Construct sequences (S), {g.) which converge uniformly on some set E, but suchthat (fag.) does not converge uniformly on E (of course, Sa9.) must converge onE).4. ConsiderFor what values of x does the series converge absolutely? On what intervals doesit converge uniformly? On what intervals does it fail to converge uniformly? Is fcontinuous wherever the series converges? Is / bounded?166 PRINCIPLES OF MATHEMATICAL ANALYSIS5.S(x) = sinShow that (f) converges to a continuous function, but not uniformly. Use theseries E fa 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, putf.(x) -1+ nxShow that (f.) converges uniformly to a function f, and that the equationS'(x) = lim f(x)is correct if x # 0, but false if x= 0.8. IГI(x) =(xS0),(x>0),if (x,) is a sequence of distinct points of (a, b), and if Elc.l converges, prove thatthe seriesS(x) =x- x.)converges uniformly, and that fis continuous for every x + x,.9. Let {fa} 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 FUNCTIONS 16710. Letting (x) denote the fractional part of the real number x (see Exercise 16, Chap. 4,for the definition), consider the functionS(x) = (nx)(x real).Find all discontinuities of f, and show that they form a countable dense set.Show that fis nevertheless Riemann-integrable on every bounded interval.11 Sunnesa ff lal ane defined on F ond

Solution: The function fnx is defined as follows: fnx=0 x<1n+1sin2πx 1n+1≤x≤1n0…

Answered: 10 Show that (S) converges to a…

10 Show that (S) 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.

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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