Department:Mathematics and Statistics
Level:Part II (yr 2)
Course Convenor:Professor JM Lindsay
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Curriculum Design: Outline Syllabusback to top
- Limits of sequences: basic results; monotonic sequences; subsequences.
- Infinite series: standard examples; comparison and ratio tests; absolute convergence; power series; Abel summation; double series.
- Limits and continuity of functions.
- Differentiation: the definition and basic results; compositions and inverse functions; differentiation of power series.
- Intermediate value theorem. Boundedness of functions continuous on a closed interval.
- Fourier series: examples, convergence theorems and applications.
Curriculum Design: Pre-requisites/Co-requisites/Exclusionsback to top
Pre-requisites:- MATH 101, MATH 115
Exclusions: MATH 210
Educational Aims: Subject Specific: Knowledge, Understanding and Skillsback to top
To introduce the basic concept of a limit, together with the derived concepts of convergent series, continuous functions and differentiation. To present the most important results connected with these concepts.
Educational Aims: General: Knowledge, Understanding and Skillsback to top
To introduce the student to mathematical analysis, the important area of mathematics concerned with the notion of a limit and its derivatives.
Learning Outcomes: Subject Specific: Knowledge, Understanding and Skillsback to top
At the end of the module students should be able to
• quote and understand the definition of a limit of a sequence or a function in its various forms;
• understand proofs using these definitions, and write simple examples of such proofs;
• demonstrate the convergence or divergence of the geometric and harmonic series and other standard series
• know and apply the basic tests for convergence of infinite series;
• calculate limits of particular functions involving products and ratios of polynomials and power series;
• understand the proofs of the intermediate value theorem and the theorem on boundedness of continuous functions, and apply these theorems;
- derive the Fourier Series of a periodic function
Learning Outcomes: General: Knowledge, Understanding and Skillsback to top
At the end of the module students should be able to understand precise mathematical definitions and reasoning and to give reasoned answers to questions within the subject area.
Assessment: Details of Assessmentback to top
Length of examination = 1 hour
5 weekly coursework assessments with solutions provided at workshops/tutorials
Curriculum Design: Select Bibliographyback to top
Bartle and D R Sherbert, Introduction to Real Analysis, Wiley, 1982.
K G Binmore, Mathematical Analysis (second edition), Cambridge University
J B Reade, An Introduction to Mathematical Analysis, Clarendon, 1986.
M Spivak, Calculus, Benjamin, 1967.
D S G Stirling, Mathematical Analysis, Ellis Horwood, 1987.
G H Hardy, Pure Mathematics (tenth edition), Cambridge University Press, 1963.
Curriculum Design: Single, Combined or Consortial Schemes to which the Module Contributesback to top
Theoretical Physics with Mathematics