c255exw02

# c255exw02 - McGLLL UNI y ERSITY FACULIIY OF SCIENQE FINAL...

• Test Prep
• 3

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.

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. ...
View Full Document

• Fall '09

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern