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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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