MA 500
Homework 7
November 6, 2013
1. (12 points) Say whether each of the following functions is (i) continuous, (ii) uniformly
continuous, and (iii) Lipschitz continuous on the given interval.1
a. f (x) = x
1
2
for = 1.5, on (0, 1).
b. f (x) = x
1
2
fo
MA 500
Final Exam Solutions
December 18, 2013
In working the exam, you may not use any books, notes, etc. Show your work on all but
the rst problem. Use the backs of the pages if you need more space, but please make clear
which problems the work belongs t
Final Exam Study Guide
Things to Know:
General guide: Know the material discussed in class and the related material covered
in the text. Know how to do problems like the homework problems and similar
problems in the text. The exam will be structured like
MA 500
1. Let a1 = 0 and an+1 =
limit.1
Homework 2
September 18, 2013
5 + 2an for n 1. Show that limn an exists and nd the
Hint: Use the approach demonstrated in Example 2.6.2.
1
2. Dene an = cos(n + n ) for n = 1, 2, . . . . Find lim supn an and lim inf
MA 500
Homework 4
October 2, 2013
1. (5 points) Construct a convergent series
n=1 an with each an > 0 and such that
an+1
1
= .
lim sup
an
n
Hint: Choose the terms so that the even terms a2k converge very rapidly to zero while the odd
terms a2k+1 converge
MA 500
Homework 6
October 30, 2013
1. (10 points) In each of the following cases, what value would you assign to f (0) to make
f continuous at x = 0?
a. f (x) = x sin(1/x) for x = 0.
b. f (x) =
tan x
for x = 0.
x
c. f (x) = x log(x2 ) for x = 0.
d. f (x)
MA 500
Homework 5
October 9, 2013
1. (5 points) Give an example of a sequence cfw_xk R2 such that limk xk exists
k=1
but limk xk does not exist.
2. (5 points)1 Find the closure of the following sets:
(b) cfw_(x, y ) R2 : xy < 1,
(a) Q,
(c) cfw_ x, sin
1
MA 500
Homework 3
September 25, 2013
These are problems 3.1.A, 3.1.C, and 3.1.E of the text.
1. Sum the series
n=1
1
.
n(n + 2)
Hint: Use the approach demonstrated in Example 3.1.3.
2. Prove that if
tk is a convergent series of nonnegative numbers and if
MA 500
Homework 1
September 11, 2013
1.1 For the following sets, nd the supremum and inmum in R. Which have a max or min?
a. A = a + a1 : a Q, a > 0 , where Q denotes the rational numbers.
b. B = a + (2a)1 : a Q, 0.1 a 5 .
c. C = cfw_xex : x IR .
Hint: Us
MA 500
Homework 9 Solutions
1. (5 points) Does
n=1
December 4, 2013
1
converge uniformly on the whole real line? Say why or why
x2 + n2
not.1
Solution: Since
1
x2 +n 2
tells us that
n=1
2. Let fn (x) =
x2
1
n2
for all x R and
1
n=1 n2
converges, the Weier
MA 500
Homework 9
1. (5 points) Does
n=1
x2
December 4, 2013
1
converge uniformly on the whole real line? Say why or why
+ n2
not.1
2. Let fn (x) =
x2
for x R and n = 0, 1, 2, . . . .
(1 + x2 )n
2
fn (x).
a. (5 points) Evaluate S (x) =
n=0
b. (5 points) F
MA 500
Homework 8
November 13, 2013
1. (5 points) Let fn (x) = xnenx for all x 0 and n 1. Show that cfw_fn converges to
zero on [0, ) pointwise but not uniformly.1
2. (5 points) Does the sequence fn (x) =
x
converge uniformly on R? 2
1 + nx2
Hint: Find m
MA 500
Mid-Term Exam Solutions
October 16, 2013
In working the exam, you may not use any books, notes, etc. Show your work on all but the rst
problem. Use the backs of the pages if you need more space, but please make clear which problems
the work belongs