Transcription of Math 312, Intro. to Real Analysis: Final Exam: Solutions
1 Math 312, Intro. to real Analysis: Final Exam: SolutionsStephen G. SimpsonFriday, May 8, 20091. True or false (3 points each).(a) For all sequences of real numbers (sn) we have lim infsn lim (b) Every bounded sequence of real numbers has at least one subsequen-tial (c) If the functionsfnare continuous on [0,1] and converge uniformlyto the functionfon [0,1], thenfis uniformly continuous on [0,1].True.(d) If the radius of convergence of a power series akxkisR, and if0< R < , then the series akxkconverges uniformly on ( R, R).False.(e) The integral of the limit is equal to the limit of the (10 points) The real number systemRhas been characterized in termsof Axioms A1 A4, M1 M4, DL, O1 O5, and the Completeness of these axioms fail for the rational number systemQ?
2 Give oneor more examples illustrating your only axiom that fails forQis the Completeness Axiom. Forexample, the set{x Q|x2<2}is bounded but has no least upperbound (10 points)(a) State the formal definition of what it means for a sequenceof realnumbers (sn) to converge to a >0 we can findNsuch that|sn s|< for alln > N.(b) In terms of your definition from part (a), prove directly thatlimn n 100 = | / n 100 |< whenevern >100 + 2/ we may takeN= 100 + 2/ (7 points) Calculate limn (32 34+38 316+ + ( 1)n32n). sum of this geometrical series is32 (1 12+14 )=32 11 ( 12)= (a) (2 points) Give an example of a bounded sequence of realnumberswith exactly two subsequential ,1,0,1,0,1, ..(b) (2 points) Give an example of a bounded sequence of real numberswith exactly five subsequential ,1,2,3,4,0,1,2,3,4.
3 (c) (4 points) Give an example of a bounded sequence of real numberswith infinitely many subsequential ,1,0,12,1,0,13,23,1,0,14,12,34,1, ..6. (10 points) Give an example of a sequence of continuous functions on [0,1]such thatfn 0 pointwise but not uniformly on [0,1]. (0) = 0,fn(1/n) = 1,fn(2/n) =fn(1) = 0, and letfn(x)be linear on the intervals [0,1/n], [1/n,2/n] and [2/n,1].7. (15 points) Prove that if |ak|is convergent then akis : Use the Triangle know that akis convergent if and only if the Cauchycriterion holds: given >0 we can findNsuch that|am+am+1+ +an|< whenevern > m > N. The Triangle Inequality tells us that|am+am+1+ +an| |am|+|am+1|+ +|an|.Thus, aksatisfies the Cauchy criterion whenever |ak| (3 points each) For each of the following series, tell whether the series isconvergent or divergent.
4 State which convergence/divergence test you areusing, and show any needed calculations.(a) geometric (b) ( 1)n by the Alternating Series Test.(c) ( 1)nn100n+ by thenth term test, becausen100n+ 1000 11006= 0.(d) (7n8n+ 1) by the Root Test, because7n8n+ 1 78<1.(e) by the Integral (15 points) For each of the following functions, say whether the functioniscontinuousand/oruniformly continuouson each of the three intervals[0,1],(0,1),(2, ).You are not required to justify or prove your answers.(a) continuous on ( , ).(b) continuous on any bounded interval; continuousbut not uniformly continuous on [a, ) for anya.(c)|x 12|+|x 3| continuous on ( , ).(d)11 atx= 1, hence not continuous on [0,1]. Contin-uous but not uniformly continuous on (0,1).]
5 Uniformly continuouson [2, ).(e) n= atx= 1, hence not continuous on [0,1]. Contin-uous but not uniformly continuous on (0,1). Undefined (divergent)forx > (10 points)(a) Prove directly thatxn 0 uniformly on the interval [ , ]. allx [ , ] we have|xn| , and 0 independently ofx.(b) Doesxn 0 uniformly on the interval ( 1,1)? Justify your , because for anynwe have 0n= 0 and1n= 1, hence we can findxsuch that 0< x <1 andxn= 1 (10 points)(a) Calculate limn 1 1sinnx havelimn 1 1sinnx dx= 1 1limn sinnx dx= 1 10dx= 0.(b) Justify your calculation for part (a) by stating an appropriate prop-erty of the functions sinnxand an applicable have sinnx 0 uniformly on [ 1,1]. The applica-ble theorem says: iffn funiformly on [a, b] andfnandfarecontinuous, then lim bafn= (15 points) For each of the following series, determine the set of allxsuchthat the series converges atx.]
6 Show any needed calculations.(a) n=02 whenever cosx= 1, ,x= n . Convergeseverywhere else, because then|cosx|<1.(b) n= for 1< x 1, converges everywhere else.(c) n=0(2 + n) for 1< x <1, diverges everywhere else.(d) n=0(10x)nn2+ for 1/10 x 1/10, diverges everywhere (e) n=0n(x 2) for 1< x <3, diverges everywhere (10 points)(a) Determine the coefficientsakfork= 0,1,2, ..such that k=0akxk=x2+ 1x 1.(1) havex2+ 1x 1=(x+ 1)(x 1) + 2x 1= 1+x 21 x= 1+x 2(1+x+x2+ ).Thusa0=a1= 1, andak= 2 for allk 2.(b) Over what interval is the above equation (1) valid? equation is valid over the interval ( 1,1).5
