Get premium membership and access revision papers, questions with answers as well as video lessons.
Math 313: Real Analysis Ii Question Paper
Math 313: Real Analysis Ii
Course:Bachelor Of Science (General) & Education Science/ Education Arts.
Institution: Chuka University question papers
Exam Year:2013
CHUKA
UNIVERSITY
UNIVERSITY EXAMINATIONS
THIRD YEAR EXAMINATIONS FOR THE AWARD OF DEGREE IN BACHELOR OF SCIENCE (GENERAL) & EDUCATION SCIENCE/ EDUCATION ARTS.
MATH 313: REAL ANALYSIS II
STREAMS: BSC (GEN), BED SC & ARTS Y3S2 TIME: 2 HOURS
DAY/DATE: FRIDAY 19/4/2013 11.30 AM – 1.30 PM
INSTRUCTIONS:
Answer question ONE (compulsory) and any other two questions
Adhere to the instructions on the answer booklet
Do not write on the question paper.
QUESTION ONE (COMPULSORY) (30 MARKS)
(a) (i) State the Cauchy criterion for convergence of a series. [1 Mark]
(ii) Prove that if the series ?_neN¦x_n is convergent, then ?(x?_n)?0 as n?8. Using a
counter example, illustrate that the converse of this is not true. [3 Marks]
(iii) Let (x_n ),(y_n ) and (Z_n) be sequences of real numbers such that
(x_n )=(Z_n)=?(y?_n) ?n=? (? is a pxed integer). let ?(x?_n),?(y?_n) both converge to the same limit sayl . Show that Zn?l as n?8.
[3 Marks]
(b) Let ?_n?N¦a_n be a series of positive terms. Let exist say l. Show that if
l<1, then the series is convergent. [4 Marks]
(c) (i) State the intermediate mean value theorem for a continuous function.
[2 Marks]
(ii) Let f: [a,b]?R be continuous. Prove that the range of f i.e f(x) is a bounded
closed interval. [3 Marks]
(d) Distinguish the following terms as used in Real Analysis.
(i) Riemann sum and Riemann – stieltjes sum of the function f. [4 Marks]
(ii) A step-function and a periodic function. [2 Marks]
(iii) An odd function and an even function. [2 Marks]
(iv) An exponential function and a logarithmic function. Illustrate this graphically
using functions f(x)and g(x)with a>1 and 0<a<1 respectively.
[6 Marks]
QUESTION TWO (20 MARKS)
(a) Define the following terms as used in Riemann Integral.
(i) A partition [1 Mark]
(ii) The norm of a partition P. [2 Marks]
(iii) Riemann Integrable function on a closed interval.[a,b]. [4 Marks]
(b) Show that the function f(x)=x is Riemann Integrable in [0,1] and that ?_0^1¦?f=1/2?
[7 Marks]
(c) (i) Distinguish a characteristic function and divichlet function. [2 Marks]
(ii) Hence prove that the dirichlet function on [a,b] is not Riemann Integrable.
[4 Marks]
QUESTION THREE (20 MARKS)
(a) Define a tagged partition of [a,b] [2 Marks]
(b) Let f,? be functions on [a,b]. Define the Riemann-stieltjes integral of f with respect to
?. [3 Marks]
(c) Let f(x)=x for a=x=b and define ?on [a,b]by ?x=0 for a=x< b with ?(b)=c. If (p ,t) is tagged partition of [a , b] with p={x_0,x_1,….x_n } and
define S(P,t,f,?)=t_n c. Show that ?_a^b¦?xd?(x)=bc.? [5 Marks]
(d) Prove that if,
(i) f(x) is even on the interval (-l,l), then
?_l^l¦?f(x)d(x)=2?_0^l¦?f(x)dx.?? [4 Marks]
(ii) f(x) is odd, then
?_(-l)^l¦?f(x)dx=0? [3 Marks]
(iii) f(x) is even g(x) is odd, then their products is odd. [3 Marks]
QUESTION FOUR (20 MARKS)
(a) When is an infinite series said to be uniformly convergent in an interval I? [2 Marks]
(b) If f(x) is integrable and its series is uniformly convergent, write down the general equation of its fourier series. Thus illustrate how each of its coefficients can be obtained.
[8 Marks]
(c) Find the fourier series of the function given by
f(x)= 0 -p<x<0
x 0<x<p [10 Marks]
QUESTION FIVE (20 MARKS)
(a) When is a series said to be absolutely convergent. [1 Mark]
(b) Prove that if the series ?_n?N¦Zn is absolutely convergent, then it is convergent. Using a
counter example, illustrate that the converse need not be true. [6 Marks]
(c) Let ?_n?N¦a_n be a series of positive terms.
Let lim-(n?8)??a_(n+1)/a_n ? exist say l. Prove that
(i) If l<1, then the series is convergent. [6 Marks]
(ii) If l>1, the series is convergent. [4 Marks]
(iii) If l=1, the test fails. [3 Marks]
---------------------------------------------------------------------------------------------------------------------
More Question Papers
Popular Exams
Return to Question Papers