Math 25: Advanced Calculus, Fall 2015

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Announcement: Final

Course grades have been submitted. Final solutions are here.

The Final exam will be in class:

• Final: Thursday, Dec 10, 1:00–3:00 p.m., Wellman 230.

It will cover all the material from this quarter, except that double series will not be on the final.

Some sample final questions are here.

Solutions to the sample final questions are here.

Important Dates

• Instruction begins: Friday, September 25
• Last day to add: Friday, October 9
• Last day to drop: Wednesday, October 21
• Last class: Friday, December 4
• Academic holidays: Wednesday, November 11; Thursday–Friday, November 26–27

Exams

There will be two Midterms and a Final

• Midterm 1: Wednesday, Oct 21 (in class)
• Midterm 2: Wednesday, Nov 18 (in class)
• Final: Thursday, Dec 10, 1:00–3:00 p.m. (Exam Code R)

Midterm 1

Solutions to the midterm are here.

Midterm 1 is in class on Wednesday, October 21st. It will be closed book. An outline of the topics is as follows:

• Proofs, including proof by induction
• Functions and sets
• Algebraic and order properties of the real numbers
• Definition of the supremum and infimum and their properties
• Completeness axiom of the real numbers
• Sequences, convergence, and limits
• Properties of limits
• Convergence of monotone sequences
• Definition of the limsup and liminf of a bounded sequence
• Relations between the limsup, liminf, and the limit of a sequence

These topics are covered in Sections 1.1–1.4, 2.1–2.4 of the text and Sections 2.1–2.7, 3.1–3.6 of the course lecture notes.

Some sample midterm questions are here. Solutions to the sample midterm questions are here.

Midterm 2

Solutions to Midterm 2 are here.

Midterm 2 is in class on Wednesday, November 18. It will cover sequences (starting with the material since Midterm 1) and series. Topology will not be on the exam. An outline of the topics is as follows:

• Cauchy sequences
• Subsequences
• The Bolzano-Weierstrass theorem
• Convergence of series
• Geometric, telescoping, and p-series
• Cauchy condition for series
• Absolutely and conditionally convergent series
• The comparison test
• The alternating series test
• Rearrangements of series
• Double series as iterated sums

These topics are covered in Sections 2.5–2.8 of the text and in the course lecture notes (with more material than we covered in the lectures).

Some sample midterm questions are here. Solutions to the sample midterm questions are here.

Grade

Grade will based on exams and homework, weighted as follows:

• 15%: Homework
• 25%: Each Midterm
• 35%: Final

Text

Understanding Analysis, Stephen Abbott, Second Edition, 2015.

This class will cover the first three chapters of the text:

• Chapter 1: The Real Numbers
• Chapter 2: Sequences and Series
• Chapter 3: Topology of the Real Numbers

The department syllabus and list of topics for the class is here.

Lecture Notes

In addition to the text, I will use my own lecture notes available online here:

An Introduction to Real Analysis

Background material is in:

• Chapter 1: Sets and Functions

The class will cover Chapters 2–5 of the notes:

• Chapter 2: Numbers
• Chapter 3: Sequences
• Chapter 4: Series
• Chapter 5: Topology of the Real Numbers

Homework

Homework will be assigned weekly and a hard copy will be due in class on Friday. Please write clearly, or type, and staple your solutions.

In addition, please put your name and section number on your homework.

The standard tool for writing mathematical papers is LaTeX, or one of its many variants. If you want to learn how to use it, see here to get started. (This suggestion is only if you're interested --- you're not required to use latex for your homework.)

Problem numbers refer to the exercises in the text (2nd edition).

• Set 1 (Fri, Oct 2)
  Sec 1.2, p. 11: 1.2.1, 1.2.3, 1.2.4, 1.2.5, 1.2.6, 1.2.8, 1.2.9, 1.2.10, 1.2.11, 1.2.12

  Set 2 (Fri, Oct 9)
  Sec 1.3, p. 18: 1.3.1, 1.3.2, 1.3.3, 1.3.4, 1.3.5, 1.3.8, 1.3.9, 1.3.11
  Sec 1.4, p. 24: 1.4.2, 1.4.4

  Set 3 (Fri, Oct 16)
  Sec 2.2, p. 47: 2.2.2, 2.2.4,
  Sec 2.3, p. 54: 2.3.1, 2.3.3, 2.3.6, 2.3.10
  Sec 2.4, p. 59: 2.4.1, 2.4.2, 2.4.3, 2.4.7

  Set 4 (Fri, Oct 30)
  Sec 2.5, p. 65: 2.5.1, 2.5.2, 2.5.4, 2.5.5, 2.5.8, 2.5.9
  Sec 2.6, p. 70: 2.6.2, 2.6.4, 2.6.7

  Set 5 (Fri, Nov 6)
  Sec 2.4, p. 59: 2.4.8, 2.4.9
  Sec 2.7, p. 76: 2.7.2, 2.7.4, 2.7.7, 2.7.8, 2.7.9, 2.7.12, 2.7.13

  Set 6 (Fri, Nov 13)
  Sec 2.7, p. 76: 2.7.1
  Sec 2.8, p. 80: 2.8.1, 2.8.2, 2.8.3, 2.8.4, 2.8.5, 2.8.6, 2.8.7

  Set 7 (Fri, Dec 4)
  Sec 3.2, p. 93: 3.2.2, 3.2.3, 3.2.4, 3.2.6, 3.2.8, 3.2.14, 3.2.15
  Sec 3.3, p. 99: 3.3.1, 3.3.2, 3.3.5, 3.3.6, 3.3.8, 3.3.11