Homework 8 Math CS 120, Winter 2009
Due on Thursday, March 12th, 2009
1. Investigate the convergence of
an where
(a)
an =
(b)
n+1
n+1
n
.
n+1 n
.
n+1
an =
2. Let
an and
bn converge, with bn > 0 for all n. Suppose that
an /bn L. Prove that
n=N an
n=N bn
li
CCS Math 120: Calculus on Manifolds
Simon RubinsteinSalzedo
Spring 2004
0.1
Introduction
These notes are based on a course on calculus on manifolds I took from Professor
Martin Scharlemann in the Spring of 2004. The course was designed for rst-year CCS
ma
Daisy He
August 29th, 2013
Do not judge people with their appearance. People have been learning this from their
parents, schools, friends, and life experences thoughout their life, but only one out of ten can
actually follow this rule. Unfortunately, I am
Identifying Rhetorical Strategies: Logos, Pathos, and Ethos
Rhetoric: The art of using language persuasively and effectively
Logos = Logic
Pathos = Emotion
The use of logic, rationality, and critical reasoning to
persuade. Logos appeals to the mind. Logos
Review Material
For ACCTG 10
Final Exam
Accounting is called the language of business; therefore it is essential that you are to match
the following terms with their definitions on the exam.
Account numbers
Accounting cycle
Accounts Receivable
Accrued wag
Homework 8 Math CS 120, Winter 2009
Due on Thursday, March 12th, 2009
1. Investigate the convergence of
an where
(a)
an =
(b)
n+1
n+1
n
.
n+1 n
.
n+1
an =
2. Let
an and
bn converge, with bn > 0 for all n. Suppose that
an /bn L. Prove that
n=N an
n=N bn
li
Homework 7 Math CS 120, Winter 2009
Due on Thursday, February 26th, 2009
1. Let f : [a, b] R be a continuous function, dierentiable in (a, b).
Prove that if f (x) = 0 for all x (a, b), then f is a constant.
2. Let f : [a, b] R be a continuous function, di
Homework 6 Math CS 120, Winter 2009
Due on Thursday, February 19th, 2009
1. Given A Rn , and z Rn , prove that the following statements are
equivalent:
(a) z A (the closure of A, also denoted by cl(A).
(b) There exists a sequence cfw_xn nN A such that xn
Homework 1 Math CS 120, Winter 2009
Due on Thursday, January 15th, 2009
1. Prove that the function
x
: Rn R dened by
= max |xi |
1in
is a norm.
2. Prove that the function
: Rn R dened by
1
n
x
|xi |
=
1
i=1
is a norm.
3. The objective of this problem is t
Homework 2 Math CS 120, Winter 2009
Due on Thursday, January 22nd, 2009
1. Prove that a convergent sequence cannot converge to two dierent
limits.
2. Consider the sequence of real numbers
an = n + 1 n.
Prove that it converges (and nd the limit), using the
Homework 3 Math CS 120, Winter 2009
Due on Thursday, January 29th, 2009
1. Consider a sequence of real numbers cfw_xn nN R. Prove that if there
exists a subsequence of cfw_xn nN that converges to b R, then
lim inf xn b lim sup xn .
n
n
2. Find
(a)
lim
x0
Homework 3 Math CS 120, Winter 2009
Due on Thursday, January 29th, 2009
1. Consider a sequence of real numbers cfw_xn nN R. Prove that if there
exists a subsequence of cfw_xn nN that converges to b R, then
lim inf xn b lim sup xn .
n
n
2. Find
(a)
lim
x0
Homework 4 Math CS 120, Winter 2009
Due on Thursday, February 5th, 2009
1. Consider the Fibonacci sequence a1 = a2 = 1, and
an+1 = an + an1 , n 2.
Prove that
an+1
n an
lim
exists, and nd the limit.
2. Let x1 = c, and for n 2 dene the sequence cfw_xn by
7
Homework 5 Math CS 120, Winter 2009
Due on Thursday, February 12th, 2009
1. Consider Rn , and the norms
p,
for 1 p < .
(a) Find constants cp > 0, dp > 0 such that
cp x
p
x
dp x p , x Rn .
(b) For any 1 p, q , nd constants cp,q > 0, dp,q > 0 such that
cp,