Trent University, Winter 2012

Mathematics 3790H

*Analysis I: Introduction to analysis*

(Formerly Mathematics 309H.)

MATH 3790H is an introductory course about analysis, with an emphasis on the concepts underpinning the convergence of series and the foundations of calculus.

**Prerequisite:** Mathematics 1100 or 1101Y, with at least 60%.

**Pre- or co-requisite:** Mathematics 2200H, with at least 60%.

**Exclusions:** Mathematics 206H and 309H.

Instructor | Text | Marking | Meetings | Content | Honour & Help | Aids | Handouts & Work | Other sources

Exam Period Office Hours

Tuesday, 10 April, 11:00-13:00

Friday, 13 April, 11:00-13:00

~~Monday, 16~~ Tuesday, 17 April, 11:00-13:00

Wednesday, 18 April, 11:00-13:00

Friday, 20 April, 10:00-12:00

Pre-Exam Study Tea

Thursday, 10:00-14:00, in GCS 338.

Stefan Bilaniuk

*office:* GCS 337

*hours:* Monday 12:00-12:50, Tuesday 13:00-13:50, Wednesday 11:00-11:50, and Thursday 10:00-10:50, or by appointment, or just drop by!

*phone:* 705 748-1011 x7474

*home phone:* 705 742-7862 [Do *not* call between 10 p.m. and 8 a.m. unless it's an emergency.]

*e-mail:* `sbilaniuk{at}trentu{dot}ca`

and `stefanbilaniuk{at}cogeco{dot}com`

[E-mail sent to my Trent address occasionally just vanishes. If it's important, send it to both.]

*web:* `http://www.trentu.ca/mathematics/sb/`

Elementary Real Analysis (Second Edition), by B.S. Thomson, J.B. Bruckner, and A.M. Bruckner,

`classicalrealanalysis.com`, 2008, ISBN 1-434841-61-8.

Parts of the text, including the sections we need for the first half of the course, may be downloaded for free in pdf format from the given web site. The entire text may be ordered in paperback form from Amazon.

*Lectures:*Monday 14:00-14:50 in BL 402, Wednesday 12:00-12:50 and Thursday 11:00-11:50 in CC K1.*Tutorial:*Thursday 125:00-12:50 in CC K1.

There will be at least ten weekly quizzes, at least ten weekly assignments, and a take-home final examination. Quizzes will normally be written weekly in the ~~Thursday~~ Monday lectures and last between ten and twenty minutes each. The final examination will be handed out two weeks before the end of classes ad will be due near the end of the examination period in April. (Please consult the handout Readings and Schedule for a detailed list of dates.) The work will weigh as follows in the final mark:

Best 9 quizzes (2% each) | 18% |

Best 9 assignments (5% each) | 45% |

Final Examination | 37% |

Assignments will not normally be accepted after the due date. Students who miss more than one quiz or assignment for reasons beyond their control should contact the instructor as soon as possible. Note that there is no attendance requirement *per se,* but the consequences of missing class are ultimately the students' responsibility to deal with.

This scheme may be modified for students in *exceptional* circumstances, such as a lengthy absence due to illness. Any such modification will require the agreement of both the student and the instructor.

MATH 3790H is a introductory course about analysis, with an emphasis on series. We will cover parts of Chapters 1 and 2 and most of Chapters 3, 5, 9, and 10 of the text, along with with a few necessary bits and pieces from other chapters. In particular, we will work through the following:

- Order properties of the real numbers (Sections 1.4-1.7)
- Limits and convergence of sequences (most of Chapter 2)
- ums and convergence of series (most of Chapter 3)
- Limits and continuity (most of Chapter 5)
- Sequences and series of functions (most of Chapter 9)
- Power series (Section 7.12, most of Chapter 10)

The obligatory statement concerning **academic integrity** reads as follows:

Academic dishonesty, which includes plagiarism and cheating, is an extremely serious academic offence and carries penalties varying from a 0 grade on an assignment to expulsion from the University. Definitions, penalties, and procedures for dealing with plagiarism and cheating are set out in Trent University's Academic Integrity Policy. You have a responsibility to educate yourself - unfamiliarity with the policy is not an excuse. You are strongly encouraged to visit Trent's Academic Integrity website to learn more -www.trentu.ca/academicintegrity.

For clarity, the following guidelines will apply in MATH 3790H:

Students are permitted and encouraged to study together and to work together on the assignments, consult any books or other sources you wish, and ask anyone willing (especially the instructor!) for hints, suggestions, and help. However,

students must write up all work submitted for credit entirely by themselves, giving due credit to all relevant sources of help and information. No aid may be given or received on the quizzes and final exam, except with the intructor's permission.

In some circumstances students may also be eligible for special help or accommodation. The obligatory statement concerning **access to instruction** reads as follows:

It is Trent University's intent to create an inclusive learning environment. If a student has a disability and/or health consideration and feels that he/she may need accommodations to succeed in this course, the student should contact the Disability Services Office (Blackburn Hall Suite 132, 705 748-1281,disabilityservices{at}trentu{dot}ca) as soon as possible. Complete text can be found under Access to Instruction in the Academic Calendar.

Students may use whatever calculators they wish. Symbolic computation software such as `Maple`

or `Mathematica`

may also come in handy when doing some of the assignments or to check your answers when studying. For the quizzes students may also bring an 8.5" by 11" aid sheet, with whatever they want on written on all sides of it.

"Personal response systems" such as clickers will not be used in MATH 3790H and it will make only minimal use of myLearningSystem/WebCT.

The last date to drop Winter half-courses without academic penalty is Friday, 9 March, 2012.

- MATH 3790H Course Outline
- Readings and Schedule
- Assignment # 1,
2,
3,
4,
5,
6,
7,
8,
9,
10

and Solutions to Assignment # 1, 2, 3, 4, 5, 6, 7 - Quizzes and Quiz Solutions
- Take-Home Final Examination
- Gauss's Test

From the online Resources for A Radical Approach to Real Analysis (2nd edition) by David Bressoud.

Stuff from previous years:

**MATH 3790H, Fall 2010**- Assignment # 1,
2,
3,
4,
5,
6

Solutions to Assignment # 1, 2, 3, 4, 5 - Quizzes and Solutions to the Quizzes
- Take-Home Final Examination
- Gauss's Test

From the online Resources for A Radical Approach to Real Analysis (2nd edition) by David Bressoud.

- Assignment # 1,
2,
3,
4,
5,
6
**MATH 3790H, Fall 2009**- Assignment # 1,
2,
3,
4,
5,
6

Solutions to Assignment # 1, 2, 3, 4, 5, 6 - Quizzes and Solutions to the Quizzes
- Take-Home Final Exam
- Archimedes' summation of a series

An excerpt from his Quadrature of the Parabola, taken from The Works of Archimedes, edited by T.L. Heath, available at the Internet Archive. - Gauss's Test

From the online Resources for A Radical Approach to Real Analysis (2nd edition) by David Bressoud.

- Assignment # 1,
2,
3,
4,
5,
6
**MATH 3790H, Fall 2008**

There is fair bit of material available online about real analysis and related areas; a brief selection follows:

- Analysis of Functions of a Single Variable by Lawrence W. Baggett.
- A ProblemText in Advanced Calculus by John M. Erdman.
- A first Analysis course by John O'Connor.
- Linear Partial Differential Equations and Fourier Theory by Marcus Pivato.
- Real Analysis (Second Edition), by Brian S. Thomson, Judith B. Bruckner, and Andrew M. Bruckner.
- Interactive Real Analysis by Bert G. Wachsmuth.
- Analysis WebNotes by John Lindsay Orr.
- Mathematical Analysis, Volume I, by Elias Zakon.

A lot of useful background can be found in http://www.trillia.com/zakon1.html by Elias Zakon.

- Real analysis at The Math Forum
- Resources for A Radical Approach to Real Analysis (2nd edition) [a text formerly used in this course], by David E. Bressoud.
- Real analysis at Wikipedia
- Analysis at Wolfram MathWorld

Department of Mathematics Trent University

Maintained by Stefan Bilaniuk. Last updated 2012.04.12.