Analysis Spring 16 - Complement to the Infinite series
chapter
Lise-Marie Imbert-Gerard
March 23, 2016
What we call comparison test for series is any test to conclude
P that a series
converges by using a comparison |an | bn with a converging series
bn .
1
MATH-UA.325.005: Analysis 1.
Homework 6
10/12/13
Solutions to be returned by the beginning of recitation on Friday, 10/18.
1. Decide which of the following statements are true and which are false. Prove the
true ones and provide counterexamples for the fa
MATH-UA.325.005: Analysis 1.
Homework 9
11/07/13
Solutions to be returned by the beginning of recitation on Friday, 11/15.
1. Assume that (ex ) = ex . Decide which of the following statements are true and
which are false. Prove the true ones and provide c
MATH-UA.325.005: Analysis 1.
Homework 7
10/31/13
Solutions to be returned by the beginning of recitation on Friday, 11/8.
1. Suppose that I is an open interval containing a and f, g : I R are in C (I).
Decide which of the following statements are true and
MATH-UA.325.005: Analysis 1.
Homework 7
10/25/13
Solutions to be returned by the beginning of recitation on Friday, 11/1.
1. (a)
Let I be a bounded interval. Prove that if f : I R is uniformly
continuous on I , then f is bounded on I . (Hint: try to use T
MATH-UA.325.005: Analysis 1.
Homework 4
9/26/13
Solutions to be returned by the beginning of recitation on Friday, 10/4.
1. Find the limit (if it exists) of the following sequences.
(1 mark) xn = 2n/(n2 + 7)
(1 mark) xn = (n3 + n2 + n)/(2n3 + n 2)
2. (3
Analysis I, section 5 - Final Exam (12/17/2013) [100 marks]
Name:
Please justify all your answers, unless otherwise instructed. No calculators allowed.
Duration: 100 minutes.
1. Provide precise denition for the following objects (3 marks each):
Convergen
MATH-UA.325.005: Analysis 1.
Homework 6
Solutions
1. Decide which of the following statements are true and which are false. Prove the
true ones and provide counterexamples for the false ones.
(1 mark) If f is continuous on (a, b) and real valued on [a, b
MATH-UA.325.005: Analysis 1.
Homework 10
11/14/13
Solutions to be returned by the beginning of recitation on Friday, 11/22. You can use
only denitions and those theorems that we proved on class.
1. (2
marks)
Prove that
1
xdx =
0
1
2
Hint: try to use the e
Topics to Review for Final Exam
From the Past:
1) Understand how to solve the basic labor supply model (e.g. like the
problem on the First Midterm), possibly with a different utility function (e.g.
like on the practice problem sets). Understand how the
Econ-UA 353 HW #3 Solution
Question 1.
(1) The employees expected utility from choosing eH is 0.8 2500 + 0.2 0
16 = 24, while her expected utility from choosing eL is 0.4 2500+0.6 00 =
20. Choosing eH results in a higher expected utility, therefore under
Lecture 22 Announcements
Today
- Finish Human Capital (just paper)
- Start Inequality
(Chapter 15)
Reminder Stata Tutorial tomorrow night 7:00-9:00 PM.
Room TBD, will email you as soon as I know
Human Capital
Recall the very broad definition of human c
Lecture 19 Announcements
Today: Part 2 of Human Capital (Chapter 9)
Goals from Last Lecture
1) Definition of Human Capital (Beckers Quote), parallels to Physical
Capital
2) Know basic costs and benefits associated with acquiring human
capital.
3) Mode
Lecture 16 Announcements
Today: Talk Contracts / Incentives (Chapter 11)
Todays lecture will not be on Midterm 2, will be on Final
Exam
Reminder: The second midterm will take place on
Wednesday, April 6. Will have a review session on
Monday night, 7:30
Lecture 23 Announcements
Today
- Continue Inequality
(Chapter 15)
Will Collect Response Papers for Autor Levy Murnane at
the end of class.
Goals
1) Recall three motivating facts from the Data in Piketty
2) Multiple Measures of Inequality each may be t
Lecture 17 Announcements
Today: Talk Contracts / Incentives (Chapter 11)
Reminder: The second midterm will take place on
Wednesday, April 6. Will have a review session
TONIGHT, 7:30-9:30, in Silver 408.
Goals from Last Lecture
1) Understand what is mea
Final exam reminder:
December 21 (Wednesday), 4:00PM - 5:15PM
Same room as lectures
Econ-UA 353 Practice Questions III
Question 1.
Suppose an industry has ten firms. The market shares of each firm are: 25%,
15%, 10%, 10%, 8%, 8%, 7%, 7%, 5% and 5%.
(1) Wh
Lecture 20 Announcements
Today: Part 3 of Human Capital (Chapter 9)
Six classes left:
Today Post-Schooling Investments in Human Capital
Wednesday 4/20 Pre-Schooling Investments in Human Capital
Monday, 4/25 Some Human Capital Topics / Income Inequali
Lecture 13 Announcements
Material on the Radar:
Chapter 5: Frictions in the Labor Market
- Monopsony, Job Search
Chapter 13: Unions and the Labor Market
Chapter 11: Contracts / Incentives in Firms (Chapter 11).
Lecture 13 Roadmap / Goals
What I want
MATH-UA.325.005: Analysis 1.
Homework 11
11/27/13
Solutions to be returned by the beginning of recitation on Friday, 12/06. Note that
this HW sheet is longer than the usual one since it covers the topic of three lectures.
1. Decide which of the following
MATH-UA.325.005: Analysis 1.
Homework 3
9/19/13
Solutions to be returned by the beginning of recitation on Friday, 9/27.
1.
(3 marks) (see also Wade's Exercise 1.6.7) A real number x is called algebraic
if it is a root of some polynomial with integer coe
Analysis I, section 5 - SAMPLE Midterm (10/22/2013) [100 marks]
Name:
Please justify all your answers, unless otherwise instructed. No calculators allowed.
Duration: 75 minutes.
1. See Wades book
2. See Wades book
3. See Wades book
4. Decide which of the
Analysis I, section 5 - SAMPLE Final Exam (12/10/2013) [100 marks]
Name:
Please justify all your answers, unless otherwise instructed. No calculators allowed.
Duration: 100 minutes.
1. Provide precise denition for the following objects (3 marks each):
Co
Analysis I, section 6 - Solutions to Quiz 1 (10/04/2013) [10 points]
1. [2 points] Recall the Field Axioms (closure, associative and commutative properties for sum and product, distributive law, additive and multiplicative identities and inverses) and the
Analysis I, section 6 - Solutions to Quiz 4 (12/06/2013) [10 marks]
1. Let f (x) = 2x3 .
(a) [2 marks] Using the Fundamental Theorem of Calculus, nd
2
0
f (x)dx.
Solution:
According to the Fundamental Theorem of Calculus, if F is a dierentiable function s
Analysis I, section 6 - Solutions to Quiz 3 (11/15/2013) [10 marks]
1. Let f : R R be a function.
(a) [2 marks] State the denition of f (a), the derivative of f at a R.
Solution:
f (a) = lim
ha
f (a + h) f (a)
.
h
(b) [3 marks] For f (x) = x|x|, nd f (a)
MATH-UA.325.005: Analysis 1.
Homework 5
10/03/13
Solutions to be returned by the beginning of recitation on Friday, 10/11.
1. (3 marks) Let cfw_xn be a real sequence and r a real number. Prove that lim supn xn <
r implies xn < r for n large enough. Prove
2.2.9. a) Let E = cfw_k Z : k 0 and k 10n+1 y . Since 10n+1 y < 10, E cfw_0, 1, . . . , 9. Hence w := sup E
E . It follows that w 10n+1 y , i.e., w/10n+1 y . On the other hand, since w + 1 is not the supremum of E ,
w + 1 > 10n+1 y . Therefore, y < w/10n
HOMEWORK 2 SOLUTIONS FOR V63.0325-001: ANALYSIS 1 - SPRING 2011
DMYTRO KARABASH
1. 1.6.7.
(a) We can write q =
(1)
m
k,
m, k Z, k = 0. Then we want to nd when equation for which
x = nq
is a solutions. How is (1) not algebraic? Well nq is not an integer. S