Math 327A (Ward)
Midterm Solutions
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Instructions.
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Math 372A Quiz 1
1. Write the triangle inequality.
|a + b| |a| + |b|.
2. Let cn be a sequence converging to C . Write the precise denition of
what it means for lim cn = C . Any equivalent form to the denition in class
n
or the book is acceptable.
For any
Math 327A Homework 4 Solutions
Due Mon July 16
TRY THESE BEFORE LOOKING! Solutions begin on next page.
1
A. Page 516
1) a) The set is (, 1) which is an open interval, so the set is open.
b) The set is [0, ). The set equals its set of accumulation points,
Math 327A Homework 3 Solutions
Due Mon July 9
A. Book Problems: Page 79, problem 2:
We rst show that if m is any rational number, then m 2 is irrational.
n
n
c
Suppose not. Then there are integers c and d such that m 2 = d . Thus
n
cn
2 = dm is a rationa
Math 327A Homework 5
Due Mon July 23
A. Page 519 Problem 1: Suppose xn is a bounded sequence in which all
values are distinct. Suppose cfw_xn has exactly 1 accumulation point x. We
will prove that lim xn = x. Suppose cfw_xn [A, B].
n
Suppose for contrad
Math 327A Homework 2
Due Mon July 2
A. Page 65 problem 20.
B. Prove that lim n1/n = 1. Hint: Follow the outline of problem 17 on
n
page 65.
Proof. First lets consider an = n1/2n . For n > 1 we can write an = 1+xn
where xn is a positive number. Now by (1.2
Math 327A Homework 8 Solutions
Page 617 problem 1 part c.
1
2 2
Let fn (x) = en x . Claim: fn 0 uniformly on all of R. First, all of
n
2 2
the functions attain their max at x = 0. This is because fn (x) = 2xnen x
which only equals 0 at x = 0.
1
This shows
Math 327A Quiz 2 Solution
1. (4 Points) Prove the following: If bn = 0 is a sequence converging to
B = 0, then there is some real number M > 0 such that |bn | > M for all n.
Choose > 0 small enough so that |B | > 0.
Since |bn | converges to |B | we can nd
Math 327A Quiz 3
1. Prove the following: If an > 0 is a sequence converging to A and A = 0,
then lim an = A.
n
Fix > 0. Suppose an > 0 converges to A and A = 0. This means we
can nd an integer N such that if n N we have |an A| < A. Suppose
n N.
| an A| =
Math 327A (Ward)
Midterm (60 Minutes)
July 16, 2012
DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO BEGIN!
Instructions.
(1) Write your name on the line provided below these instructions.
(2) Your solutions must be NEAT and LEGIBLE.
(3) Show all work you want g
Math 327A (Ward)
Final (60 Minutes)
July 17, 2012
DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO BEGIN!
Instructions.
(1) Write your name on the line provided below these instructions.
(2) Your solutions must be NEAT and LEGIBLE.
(3) Show all work you want gra
Math 327A (Ward)
Final (60 Minutes)
Aug 17, 2012
DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO BEGIN!
Instructions.
(1) Write your name on the line provided below these instructions.
(2) Your solutions must be NEAT and LEGIBLE.
(3) Show all work you want grad
Math 327A Quiz 4
1. Write the denition of what it means for an to be a Cauchy sequence.
For all > 0 there exists an integer N such that if n, m N , then
|an am | < .
2. Let xn be a sequence of distinct real numbers. Dene E = cfw_L :
L is a subsequential l
Math 327A Quiz 5
The geometric series formula is 1 + r + r2 + + rk1 =
n2 + 1
converge?
1 n + 2n2
1. (2 points) Does the series
No. The limit of the terms
1 rk
.
1r
n=1
is 1 so
2
they dont go to 0.
n
1
converges by showing that its sequence of
2. (5 points
Math 327A Quiz 4
1. Write the denition of what it means for an to be a Cauchy sequence.
2. Let xn be a sequence of distinct real numbers. Dene E = cfw_L :
L is a subsequential limit of xn . Prove E is closed.
Math 327A Quiz 5
The geometric series formula is 1 + r + r2 + + rk1 =
1. (2 points) Does the series
n=1
n
1 rk
.
1r
n2 + 1
converge?
1 n + 2n2
1
converges by showing that its sequence of
3
n=1
partial sums forms a Cauchy sequence (for full credit show thi
Math 327A Quiz 2
1. (4 Points) Prove the following: If bn = 0 is a sequence converging to
B = 0, then there is some real number M > 0 such that |bn | > M for all n.
2. (6 Points) Use part 1 to prove: If bn = 0 converges to B = 0, then
1
1
lim
=.
n bn
B
Math 327A Quiz 6
1. (5 points) Use the ratio test to determine the values of x for which the
xn
series
converges absolutely.
2n
n=1
Compute
|x|n+1
n+1
lim 2|x|n
n
2n
|x|
|x|
=
.
n 2
2
= lim
The ratio test tells us this series converges absolutely when
for
Math 372A Quiz 1
1. Write the triangle inequality.
2. Let cn be a sequence converging to C . Write the precise denition of
what it means for lim cn = C . Any equivalent form to the denition in class
n
or the book is acceptable.
3. What does it mean for th