# Concepts of Real Analysis

__ Schedule of classes:__

Section 3, M W F 11:15 AM - 12:05 PM in 203 WILLARD

Section 4, M W F 10:10 AM - 11:05 AM in 106 SACKETT

__FINAL EXAM:__ MAY 2, 12:20 - 2:10 PM, 022 DEIKE
(comprehensive).

__ REVIEW SESSION__ FRIDAY APRIL 26 in class. FINAL QUIZ
WEDNESDAY APRIL 24.

__Finals Week Office Hours:__ SUNDAY 4:00-5:30 PM, WEDNESDAY 11:00
AM- 12:30 PM.

Topics for the final exam.

Review Problems for the final exam.

Practice tests with solutions available on ANGEL.

__MIDTERM 2:__ MONDAY MARCH 25 in class.

__ REVIEW SESSION__ FRIDAY MARCH 22 in class.

__ Office hours__ for the exam:
12:15 - 1:45 PM, Friday March 22. NO Office
hours, Monday March 25.

Topics for the exam.

Review Problems for the exam.

Practice Test (SKIP Problem 1, solutions available on ANGEL).

__MIDTERM 1:__ FRIDAY FEB 8 in class.

__ REVIEW SESSION__ WEDNESDAY FEB 6 in class.

Topics for the exam.

Review Problems for the exam.

Practice Test (solutions available
on ANGEL).

__ Make-up Classes__: MONDAY JANUARY 21 at the usual time and
place for each section

SUNDAY JANUARY 27, 4:00 - 5:30 PM in 101 OSMOND (both sections).

__NO classes__ MONDAY FEB 4, MONDAY FEB 18, FRIDAY FEB 22.

__Topics:__-
**Chapter 1, Real Numbers**: Review of properties of real numbers (Sections 1.1.1,1.1.2,1.2.1,1.2.2,1.5), irrational numbers (1.1.3), modulus and distance (1.2.3), triangle inequality, Bernoulli and Cauchy-Schwartz inequalities (1.3), sumpremum and infimum, completeness (1.4), Archimedean and density properties (1.1.4). -
**Chapter 2, Sequences**: monotonic sequences (2.1), null sequences (2.2), convergent and divergent sequences (2.3, 2.4), the Monotone Convergence Theorem (2.5). -
**Chapter 3, Series**: definition and examples, geometric series, telescoping series, non-null test (3.1), series of positive terms, comparison, root and ratio, Cauchy tests (3.2), alternating series, rearrangements, products (3.3), the exponential function (3.4).

**(from the textbook): every problem is worth**

__Homework Problems__**5 points**. Complete solutions available on ANGEL under "Lessons" tab.

- Section 1.1: 5, Section 1.3: 1, 7. Section 1.4: 4, 5 a), 6 (note
|a+b| should be |a-b| in the second formula of this problem), pages 35-36.
**DUE WEDNESDAY JANUARY 23.** - Section 2.1: 2,3. Section 2.2: 1, 2, 3, 7 (Hint: it is
equivalent to show that (a_n) is null if (a_n^2) is
null), pages 79-80.
**DUE WEDNESDAY JANUARY 30.** - Section 2.2: 4, 5, 6. Section 2.3: 1, page 80.
**DUE WEDNESDAY FEBRUARY 6.** - Section 2.3: 2, 3 (you MUST use the
__definition__of converegent sequences for this problem), page 80-81.**DUE WEDNESDAY FEBRUARY 13**. - Section 2.4: 1, 4 a), b), c), d), e), page 81 (note: a
*divergent*sequence is a sequence does not converge, an*unbounded*sequence is a sequence that is not bounded.)**DUE WEDNESDAY FEBRUARY 20.** - Section 2.4: 2, 3, 5 (
__Hint__: consider first the case of eventually monotonic sequences) page 81.**DUE WEDNESDAY FEBRUARY 27.** - Section 2.5: 1, 2, 3, 6 page 81-82.
**DUE WEDNESDAY MARCH 13.** - Section 2.5: 5 page 82. Section 3.1: 1, 2 page 127.
**DUE WEDNESDAY MARCH 20.** - Section 3.1: 3, 4, page 128 (you MUST use the techniques of
Section 3.1). Section 3.2: 1 a), b), page 128.
**DUE WEDNESDAY MARCH 27.** - Section 3.2: 1 c), d), e), f), 2, 3, page 128.
**DUE WEDNESDAY APRIL 3.** - Section 3.2: 4. Section 3.3: 1, page 128.
**DUE WEDNESDAY APRIL 10.** - Section 3.3: 2, 3 (
__Hint__: use Theorem 3 on page 112), pages 128-129.**DUE WEDENSDAY APRIL 17.** - Work out the problems found here.
**DUE WEDENSDAY APRIL 24.**

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© Anna Mazzucato Last modified: Wed Apr 24 15:23:29 EDT 2013