Math 2123b Assignment 7 Solutions
Due: 130321 in class
Do at least 4 of the following questions. Only your best 4 answers will be counted.
1. Exercise 31.18.
Solution:
1
1
xp+1
1
.
=
p+1 0 p+1
0
For p < 0, we nd xp is not dened at x = 0 so the integral is
Math 2123, Winter 2014
Real Analysis II
Syllabus
Instructors: Rasul Shakov, MC 112, shafikov@uwo.ca (emails will be answered within 48 hours).
Oce hours: TBA.
Textbook: S. Lay, Analysis: With an Introduction to Proof (4th Edition) Pearson, Prentice Hall.
Math 2123b Bonus Assignment 3 Solutions
Logarithm and Exponential Functions from First Principles
Due in class April 4, 2013
This assignment develops the logarithm and exponential functions from rst principles, using only
material covered in this course.
Math 2123b Assignment 9 Solutions
Due: 130410 (Wednesday, April 10) in class
Do at least 4 of the following questions. Only your best 4 answers will be counted.
1. Exercise 35.12
Solution: For x = 0, we have fn (0) = 0 1, so fn (0) 0. For 0 < x 1, we have
Math 2123b Assignment 8 Solutions
Due: 130328 in class
Do at least 4 of the following questions. Only your best 4 answers will be counted.
1. Exercise 33.13
Solution:
(a) Since (an ) is a decreasing sequence of nonnegative terms, we have for each k 0:
n=0
Math 2123b Bonus Assignment 2
Due in class March 7, 2013
This assignment studies an algorithm which, given a function f , nds (approximately) points x at which
f () = 0. This may seem like an overly
x
specialized problem, but in fact it is widely applica
Math 2123b Bonus Assignment 3
Logarithm and Exponential Functions from First Principles
Due in class April 4, 2013
This assignment develops the logarithm and exponential functions from rst principles, using only
material covered in this course. For that r
Western Department of Mathematics
February 28, 2013
Math 2123b Midterm Exam
2.5 hours
Solutions
Maximum mark: 70
Write your answers, solutions, and all scrap work in booklets.
Write your name and student number on this sheet and all booklets.
Do not remov
Math 2123b Assignment 5
Due: 130225 in class
1. Use Taylor polynomials for f (x) = 1 + x about x0 = 0 to obtain an approximation
the
for 2 which is accurate to 3 decimal places,i.e. the error is less than 1/1000. (Use a
calculator for the work.) Show your
Math 2123b Assignment 5 Clarications and Hints
1. In question 1, you are to nd the least n for which the Taylor polynomial for 1 + x will give 2
with at least 1/1000 accuracy. That is, you should nd the least n for which the remainder at the
appropriate v
Math 2123b Assignment 0 Solution
Due: 130117 in class
This is a bonus warm-up assignment. Marks received for this assignment may be used towards your
total assignment mark.
Either nd a sequence (s) which has every c R as a subsequential limit, and also co
Math 2123b Bonus Assignment 2 Solutions
Due in class March 7, 2013
This assignment studies an algorithm which, given a function f , nds (approximately) points x at which
f () = 0. This may seem like an overly
x
specialized problem, but in fact it is wide
Mathematics 2123
Winter 2014
Homework 2.
Due Feb 6 in class.
1. A function f is called a contraction on a set A if there exists a constant s, 0 < s < 1, such that
|f (x) f (y)| s|x y|
for all x, y A. Show that if f is of class C 1 on the interval [a, b],
Math 2123b Assignment 4 Solutions
Due: 130214 in class
x2 x 3
xn
+
+ + .
2
3!
n!
[Hint: Start by proving Exercise 26.5(a) and then use induction on n.]
1. Prove that for n N and x > 0, ex > 1 + x +
Solution: Following the hint, we rst prove that ex > 1 +
Math 2123b Assignment 2
Due: 130131 in class
1. Suppose f : D R is continuous and f (D) is compact. Is it always true that D must
be compact? Give either a proof or a counterexample, as appropriate.
Solution: It is not always true that D must be compact.
Math 2123b Assignment 3 Solutions
Due: 130207 in class
1. Prove directly from Denition 23.1 that f : R R with f (x) =
continuous. That is, give an proof.
x2
1
is uniformly
+1
Solution: Let > 0 be given. We examine
(y 2 + 1) (x2 + 1)
x+y
1
1
2
=
= |x y|
.
Mathematics 2123
Winter 2014
Homework 1.
Due Jan 23 in class.
1. Let f be a continuous function on the closed interval [0, 1] with range also contained in [0, 1]. Prove
that f must have a xed point; that is, show that f (x) = x for at least one value of x
Mathematics 2123
Winter 2014
Homework 6.
Due April 3 in class.
1. The Taylor series expansion of the function dened by f (x) = (1 + x) about x = 0 for any xed
real number is called the binomial series.
(a) Show that
( 1)( 2) . . . ( n + 1) n
x .
n!
(1 + x
Math 2123b Assignment 6 Solutions
Due: 130314 in class
Do at least 4 of the following questions. Only your best 4 answers will be counted.
1. Exercise 30.7. [Hint: The function g required may be a very simple monotone function.
(But not continuous, of cou
Math 2123b Assignment 1 Solutions
Due: 130124 in class
In this assignment limits may exist in the extended sense, i.e. may be or . Your
answers should recognize these cases when appropriate.
x2
if it exists, and show that your answer is correct.
x4
Soluti
Math 2123b Assignment 5 Solutions
Due: 130225 in class
1. Use the Taylor polynomials for f (x) = 1 + x about x0 = 0 to obtain an approximation for 2
which is accurate to 3 decimal places, i.e. the error is less than 1/1000. (Use a calculator for the
work.
Math 2123b Assignment 7
Due: 130321 in class
Do at least 4 of the following questions. Only your best 4 answers will be counted.
1. Exercise 31.18.
2. Exercise 33.12.
3. (a) Suppose that
an is absolutely convergent and that (bn ) is a bounded sequence.
Pr
Math 2123b Assignment 6
Due: 130314 in class
Do at least 4 of the following questions. Only your best 4 answers will be counted.
1. Exercise 30.7. [Hint: The function g required may be a very simple monotone function.
(But not continuous, of course.)]
2.
Math 2123b Assignment 9
Due: 130410 (Wednesday, April 10) in class
Do at least 4 of the following questions. Only your best 4 answers will be counted.
1. Exercise 35.12
2. Exercise 35.15, parts (b) and (c). You may use part (a).
3. Exercise 36.6
4. Exerci
Math 2123b Assignment 8
Due: 130328 in class
Do at least 4 of the following questions. Only your best 4 answers will be counted.
1. Exercise 33.13
2. Exercise 34.7
3. (a) Prove that: if (an ) and (bn ) are sequences of real numbers satisfying
i. an an+1 >
Math 2123b Assignment 2
Due: 130131 in class
1. Suppose f : D R is continuous and f (D) is compact. Is it always true that D must
be compact? Give either a proof or a counterexample, as appropriate.
2. Given that f : [a, b] R is continuous, use the IVT to
Math 2123b Assignment 4
Due: 130214 in class
xn
x2 x 3
+
+ + .
2
3!
n!
[Hint: Start by proving Exercise 26.5(a) and then use induction on n.]
1. Prove that for n N and x > 0, ex > 1 + x +
2. Exercise 26.14.
3. Exercise 26.15.
4. In Exercise 22.13(b) on pa
Mathematics 2123
Winter 2014
Homework X.
Due April 7.
1. Show that the integral
1
sin x
dx
x
converges for all > 0.
2. Suppose that cfw_fn converges uniformly on the intervals [a, b] and [b, c]. Prove that cfw_fn converges
uniformly on [a, c].
3. Suppos
Mathematics 2123
Winter 2014
Homework 5.
Due March 20 in class.
1. Find a formula for the sum of
(i)
(ii)
n
n=1 n(n + 1)x on (1, 1),
n
n=1 nx /(n + 1) on (1, 1).
2. Explain why a power series can converge conditionally for at most two points.
3. Show that
Mathematics 2123
Winter 2014
Homework 4.
Due March 13 in class.
1. Show that the series
p n
n=1 n r
converges for any p R, and any r, |r| < 1.
2. Consider the series
an =
n=1
1 1
1
1
1
1
1
1
+ + 2 + 2 + 3 + 3 + 4 + 4 .
2 2 2
2
2
2
2
2
(Here a2n1 = a2n = 2