Mathematical Analysis

Course Coordinator: Dr Tony M T Chan, Grad Dip, MPhil (CUHK), PhD (CityU)

Course Developer: The Open University, UK, Course Team

This course is intended to provide a conceptual understanding of the theory of mathematical analysis. The course deals with the basic theory of analysis in real-valued functions in single variables. It provides students with a good foundation for higher-level courses in mathematical studies. The course is made up of six study units. Topics include series, continuity, limits, differentiation, integration and power series. MATH S216 is one of the compulsory middle-level courses in the BSc and BSc (Hons) in Mathematical Studies, and an elective middle-level course in the BSc (Hons) in Statistics and Decision Science.

MATH S216 forms an excluded combination with MATH S203/MATH S213.

This course is presented at 12-month intervals.

Advisory prerequisite(s)
You are advised to have already studied one of the foundation mathematics courses MATH S101, MATH S111, MATH S112, MATH S121 or MATH S122.

Aims
This course aims to:

• Introduce some basic concepts in mathematical analysis;
• Develop students' competence in series and continuity;
• Develop students' ability in handling basic skills for testing the convergence of series;
• Develop students' knowledge of the limiting processes of real valued functions of one variable;
• Develop students' rigour in writing proofs.

Contents
The course covers the following topics:

• Series — convergent series, test for convergence and its proof, non-null test and alternative test, absolute convergence
• Continuity — continuity of functions, the local rule, the glue rule and the restriction rule, the intermediate value theorem, the extreme value theorem, the inverse functions and inverse function rule
• Limits — rules for limits and one-sided limits, asymptotic behaviour of functions, uniform continuity
• Differentiation — differentiable functions and test for differentiability, rules for differentiation and proofs, Roll's theorem, local extremum theorem, the mean value theorem, L'Hôpital's rule
• Integration — Riemann integral, the fundamental theorem of calculus, inequalities for integrals, Wallis's formula, integral test, Stirling's formula for n!, Riemann's criterion for integrability and proof
• Power series — Taylor's polynomials and theorem, approximation by Taylor's theorem, convergence of power series, radius of convergence, approximate derivative using power series

Learning support
There will be six two-hour tutorials and three surgeries.

Assessment
There will be two assignments and a final examination. Students are required to submit assignments via the Online Learning Environment (OLE).

Online requirement
This course is supported by the Online Learning Environment (OLE). You can find the latest course information from the OLE. Through the OLE, you can communicate electronically with your tutor and the Course Coordinator as well as other students. To access the OLE, you will need to have access to the Internet. The use of the OLE is required for the study of this course and you can use it to submit assignments.

Equipment
Students will need access to a personal computer or other electronic devices with an Internet connection, and a television to watch the course-related programmes broadcast on Sunday mornings.

Set book(s)
There are no set books for this course.

Students with disabilities or special educational needs
The audio and visual components of this course may cause difficulties for students with hearing or vision impairments. You are encouraged to seek advice from the Course Coordinator before enrolling on the course.