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(MATH202)[2008](s)final~id-^_10473.pdf
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Math 202 (Introduction to Real Analysis) Spring 2008
Final Examination C (Duration: 120 minutes)
Directions: Thisisa closedbook exam. Works (including scratchworks)mustbe shown legibly to receive credits. Answers alone are worth very little. Calculators are allowed.
Notations: R denotes the set of all real numbers.
Problems
k2
1. (5 marks) Determine the domain (of convergence) of f(x)=.
(.2x)k . Be sure
k=1
to show work.
1
.xdx
2. (10 marks) Determine whether the improper integral converges or not. .1 sin2 x 1
.xdx
Also, determine whether the principal value integral P.V. converges or not.
.1 sin2 x
(Make sure works for eachstep are shown clearly!)
. kx .k
3. (10 marks) Prove the series of functions . converges uniformly on R.
1+ k2x2
k=1
4. (10 marks) Let {xn},{yn}be two Cauchy sequences of real numbers. Prove that
.x2 +y2 is alsoa Cauchy sequenceby checking the de.nitionof Cauchy sequence.
nn
(Do not use the theorem that asserts a sequence is a Cauchy sequence if and only if it converges. Otherwise0mark willbe given for this problem!)
5. (a) (5 marks) State Lebesgues theorem.
(b) (10 marks) For n =1,2,3,..., let fn :[0,1] [0,1] be Riemann integrable
functions. Prove that g:[0,1] R de.nedby g(0) = 0 and
. 11
g(x)= fn(x) forn=1,2,3,... and x
n+1,n
is Riemann integrable on [0,1].
6. (8 marks) Let a1,a2,a3,... R and sn be the n-th partial sum of the convergent
a1 +2a2 +3a3 + +nan
series .ak. Prove that lim =0.
n n
k=1
7. (8 marks) Let f:R R beatwice di.erentiable function suchthatf.. (x)is continu-
1 1 ous and |f..(x)|1for allx[0,1].If f.1.=0,then prove that .
.f(x)dx.
2 . 0 .24.
CEnd ofPaperC