• Week 1: Read Chapter 6 (Riemann-Stieltjes Integral),
  Do Chapter 6: 1, 2, 3, 4, 5.
• Week 2: Continue reading Chapter 6.
  Do Chapter 6: 7, 8, 10, 11, 12,
• Week 3 (MLK day): Start reading Chapter 7 (Sequences and Series)
  Do Chapter 6: 15, 16, 17, 18, 19.
• Week 4: Continue reading Chapter 7.
• Do Chapter 7: 1, 2, 3, 4, 5, 6, 7.
• Week 5 (Exam): Continue reading Chapter 7.
• Do Chapter 7: Problems 8, 10, 15, 16, 18.
• Week 6: Finish reading Chapter 7 and start reading Chapter 8 (Further topics in the theory of series)
• Do Chapter 7 Problems 20, 21,22.
• Week 7: Continue reading Chapter 8.
  Do Chapter 8 Problems 1, 2, 3, 4, 5.
• Week 8: Continue reading Chapter 8.
  Do Chapter 8 Problems 6, 7, 8, 9, 10, 11.
• Week 9: Finish reading Chapter 8.
  Problems 12, 13, 14.
• Week 10: There is no new material scheduled here. It is probably not necessary to start multi-variable calculus as this will be done in Spivak nor is it probably a good idea to start on the Lebesgue integral. Instead, this provides a bit of breathing room to review the course, get caught up if you have fallen behind, and explore some interesting applications of the material that has been covered; there are, for example, some fascinating conditionally convergent series.

Course Goals

1. We begin with integration theory. The Riemann-Stieltjes integral is presented using the theory of upper and lower sums; this will be crucial subsequently in Math 4/515 when multi-variable integration is presented. The integral as a limit of sums is to be discussed as is the relation between integration and differentiation. Integration of vector functions is introduced and functions of bounded variation are treated. Some theorems of integration are given (first mean value theorem, change of variable theorem in 1 variable, rectifiable curves, etc.). This occupies approximately 3 weeks.
2. Sequences and series of functions form the second part of the course. Uniform convergence is treated - this is the "gold standard". Continuity and integration are presented (the uniform limit of continuous functions is continuous; the integral of the limit is the limit of the integral if the convergence is uniform). Equicontinuous families and the Stone-Weierstrauss theorem are presented. This again occupies approximately 3 weeks.
3. The course concludes with Chapter 8 (Further topics in the theory of series). Power series (analytic functions), the exponential and logarithm functions, the trigonometric functions, the algebraic completeness of the complex field, Fourier series etc. This puts in context much of the material the students learned informally over the years.

Mathematics Department Undergraduate Grading Standards November 2011

• (1) Mathematics is hierarchical. A student who is given a grade of C or higher in a course must have mastery of that material that allows the possibility of succeeding in courses for which that course is a prerequisite.
• (2) Some mathematics courses are primarily concerned with techniques and applications. In such courses student success is measured by the student's ability to model , successfully apply the relevant technique, and bring the calculation to a correct conclusion. The department's 100-level courses and most calculus courses are examples in this category although these are not the only examples. Other courses such as this one are primarily concerned with theoretical structures and proof. In such courses student success is measured by the student's ability to apply the theorems and definitions in the subject, and to create proofs on his or her own using the models and ideas taught during the course. Many courses are partly hybrids incorporating both techniques and applications, and some element of theory. Some lean more toward applications, others more toward theory.

• A: Consistently chooses appropriate models, uses correct techniques, and carries calculations through to a correct answer. Able to estimate error when appropriate, and able to recognize conditions needed to apply models as appropriate.
• B: Usually chooses appropriate models and uses correct techniques, and makes few calculational errors. Able to estimate error when prompted, and able to recognize conditions needed to apply models when prompted.
• C: Makes calculations correctly or substantially correctly, but requires guidance on choosing models and technique. Able to estimate error when prompted and able to recognize conditions needed to apply models when prompted.
• D: Makes calculations correctly or substantially correctly, but unable to do modeling.
• F: Can neither choose appropriate models, or techniques, nor carry through calculations.

Rubric for pure courses:

• A: Applies the important theorems from the course. Constructs counterexamples when hypotheses are weakened. Constructs complete and coherent proofs using the definitions, ideas and theorems from the course. Applies ideas from the course to construct proofs that the student has not seen before.
• B: Applies the important theorems from the course. Constructs counterexamples when hypotheses are weakened. Constructs complete and coherent proofs using the definitions, ideas and theorems from the course.
• C: Applies the important theorems from the course when the application is direct. Constructs simple proofs using the de nitions when there are very few steps between the de nitions and the conclusions. Explains most important counterexamples.
• D: Can do some single step proofs and explain some counterexamples.
• F: Unable to do even single step proofs or correctly use de nitions.

Many courses combine pure and applied elements and the rubrics for those courses will have some combination of elements from the two rubrics above. Detailed interpretation of the rubrics depends on the content and level of the course and will be at the discretion of instructors. Whether to award grades of A+ is at the discretion of instructors.

Academic dishonesty