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Unformatted text preview: McGLLL UNI y ERSITY FACULIIY OF SCIENQE FINAL EXAMINATION MATHEMATIQS 18%255B ANALYSIS II Examiner: Professor D. Jakobson Date: Monday, April 29, 2002 Associate Examiner: Professor S. Drury. Time: 2:00 RM. - 5:00 P.M. T UCTIONS Answer all questions. Each question is worth 20 points. This exam comprises the cover and 2.pages of questions. Final Ebramination ‘ ' - April 29, 2002 189-25513 FINAL Do all the problems. Every problem is worth 20 points. Problem 1. Establish the convergence/divergence for the series whose nth term is given by ~ :3») (a points) (er/(2"). b) (7 points.) 2" -nl/(n”) and 3" ~nl/ (12"). c) (7 points.) (1-3-...-(2n—1) P 2.4.....(2n) )’ p=lmp=3i Problem 2. Determine whether the following sequences ofifimctiom converge uni- fiormly or pointwise (or neither) in the regions indicated. Determine the pointwiee limits (where they exist); are the functions wntinuous/difierentiable, (in the latter case, do the derivatives converge uniformly)? a) (7 points.) Lane fn(m)={ n: , £950 1; 2:0. - £01- 1: e [—1r41r]. b) (6 points.) fn(z) = 33/(3 + W) for a: E [0, 1]. c) (7 points.) fn(z) = e”"‘/n for z e [0, 00). Problems. Letm <m<n3<... bethenumbersthatdon’tusethedigit'fin theirdecimalenrpansion.Provethat , converges. Problem 4. Suppoeethat both the series Fe) = Z We) 0:30 and the series a r t1 0(z)= f0 mam/fin 0 new”... converge uniformly on soine interval. What can you say about the function repre- sented by the series H (z) := F(a:) + 6(3)? ' Problem 5. a) (5 points:) State the Lebesgue’s integrability criterion. b) (5 points.) Define sets ofmeasure 0 (null sets) in R. Final Examination ‘ April 29, 2002 189—2553. c) .(5 points.) Prove that if g : [a,b] 4 [c,d] is Riemann integrable on [a,b], and if f is continuous on [ad], then h(a:) := f(g(z)) is also Riemann , integrable on [a, b]. ‘ d) (5 points.) Give an example of a Riemann integrable function f on [0, 1] such that sgn(f(a:)) is not Riemann integrable. ' Problem 6. . a) (10 points.) Let a,. be a sequence of real numbers such that 2a: con- verges. Prove that 2(IanI/n) also converges. b) (10 points.)Leta1 2a; _>_ as 2 2 Obeamonotonedecreasing sequence of nonnegative numbers, and let 2:11 on converge. Prove that “Elana” = 0. ...
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