Exam 1A Math 221 Spring 2015 Name
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1. (20 points}
(a) Deﬁne liman my? : m, where (E is a, real number.
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(b) Deﬁne lilmHoo sun : —00.
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MATH 325: Quiz 10 Solutions
Friday, April 8
1. (2 points) Let D R be a subset, and let f : D ! R be a function. Write down
(precisely) what it means for f to be uniformly continuous on D.
For all e > 0, there exists d > 0 such that, for all x, y 2 D, if |
MATH 325: Quiz 7 Solutions
Friday, March 4
1. (3 points) Let (sn ) be a sequence of real numbers. Write down what it means for
(sn ) to converge to a real number L.
For all e > 0, there exists N 2 R such that for all n 2 N, if n > N, then |sn
L| < e.
2. (
MATH 325: Quiz 6 Solutions
Friday, February 26
1. (2 points) Give an example of real numbers a and b such that |a + b| = |a| + |b| as
well as an example of real numbers c and d such that |c + d| 6= |c| + |d|.
For example, a = b = 1, c =
|c + d| = 1, and |
MATH 325: Quiz 8 Solutions
Friday, March 11
1. (2 points) Let (sn ) be a sequence of real numbers. Write down what it means for
(sn ) to be Cauchy.
For all e > 0, there exists N 2 R such that for all m, n 2 N, if m > N and n > N, then
|sn sm | < e.
2. (2
MATH 325: Quiz 4 Solutions
Friday, February 5
1. (2 points) Let X and Y be sets, and let f : X ! Y be a function. Write down what it
means for f to be injective and what it means for f to be surjective.
f is injective if, for all x1 , x2 2 X, f (x1 ) = f
MATH 325: Quiz 5 Solutions
Friday, February 19
1. (10 points) Use induction to prove that
3 + 11 + 19 + + (8n
5) = 4n2
n
for all natural numbers n.
If at any point, you use circular logic (assume the statement you are trying to prove),
you will receive no
MATH 325: Quiz 3 Solutions
Friday, January 29
1. (3 points) Write down the negation of the statement For all x 2 R, there exists y 2 R
such that xy = 1.
There exists x 2 R such that for all y 2 R, xy 6= 1.
2. (2 points) Is the original statement (not the
MATH 325: Quiz 2 Solutions
Friday, January 22
A, B, C, and D are statements.
1. (3 points) What is the converse of A =) B? What is the contrapositive of A =) B?
Which of these two statements is logically equivalent to A =) B?
The converse is B =) A, and t
MATH 325: Quiz 1 Solutions
Friday, January 15
1. (2 points) What is the definition of a statement?
A statement is a sentence that is either true or false, but not both.
2. (2 points) Let A and B be statements. Write down the negation of the statement A
is
Exam 2 Math 325 Spring 2015 Name
1. (13 points) Using the: deﬁnition of an inﬁnite series ﬁnd the value of
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WWI/ﬁrm “" $763 4+? (WNW jaw/LS
" ‘ ’24 5' 2 WI, 2;
ﬂ — Yf‘qu3'13 ! Lf j
(X) ’6 ,
Z i
MATH 325: Quiz 9 Solutions
Friday, April 1
1. (2 points) Let D R be a subset, let x0 be an element of D, and let f : D ! R be a
function. Write down (precisely) what it means for f to be continuous at x0 . (Do not
use the word sequence in your definition.