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| Module Availability |
| Autumn semester. |
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| Assessment Pattern |
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Unit(s) of Assessment
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Weighting Towards Module Mark( %)
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Class Test 1
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30%
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Class Test 2
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35%
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Class Test 3
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35%
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| Module Overview |
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| Prerequisites/Co-requisites |
None. |
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| Module Aims |
The objective of this module is to provide an introduction to analysis, which is the branch of mathematics that rigorously studies functions, continuity and limit processes, such as differentiation and integration. The module intends to lead to a deeper understanding of what it means when a sequence or series is said to converge. Historic motivation and the rigorous use of definitions and logic play a central role. Tools such as convergence tests are presented and their validity proved. |
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| Learning Outcomes |
At the end of the module a student should be able to:
· demonstrate understanding of the real numbers, their axioms and the role of completeness in the existence of limits
· calculate limits of sequences and (power) series, and prove/disprove converge using the definitions.
· properly interpret and apply quantifiers in mathematical statement.
· quote and apply basic theorems in analysis, notably the Theorem of Bolzano-Weierstrass and convergence tests.
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| Module Content |
- The axioms of real numbers. Denseness of rational and irrational numbers. The least upper bound property and consequence to existence of limits. The triangle inequality.
- Concept of proofs, proof by natural induction, set theoretic notation and quantifiers.
- Sequences and convergence, and their properties. Boundedness, subsequences and the Theorem of Bolzano-Weierstrass.
- Infinite series, convergence and absolute convergence. Convergence tests. Power series and radius of convergence. Rearrangement of series.
- Historical motivation of analysis in mathematics.
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| Methods of Teaching/Learning |
Teaching is by lectures, tutorials and tests. Learning takes place through lectures, tutorials, tests, exercises and background reading.
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| Selected Texts/Journals |
Recommended
J. M. Howie, Real Analysis, Springer (2001) 2nd edition, Available in paperback from Springer or Amazon.co.uk
R.P. Burn, Numbers and Functions, Steps into Analysis, Second Edition,
Cambridge
University Press (2000).
Available in paperback from CUP or Amazon.co.uk
Supplementary Reference Texts (all available in the library)
P.E. Kopp, Analysis,
Arnold Publishers, (1990).
K.E. Hirst, Numbers Sequences and Series,
Arnold Publishers, (1995).
C. McGregor, J. Nimmo and W. Stothers, Fundamentals of University Mathematics,
Albion Publishers, (1994).
M. Spivak, Calculus W.A. Benjamin (1967).
S. Lang, Analysis I Addison-Wesley (1968). |
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| Last Updated |
13 October 2008 |
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