Registry
Module Specifications
Archived Version 2023 - 2024
| |||||||||||||||||||||||||||||||||||||
Description This module introduces students to the formal and rigorous approach to mathematics which underpins mathematical analysis. The students will develop the skills necessary to make the transition from a formulaic understanding of mathematics to constructing their own formal mathematical arguments, and to promote advanced mathematical thinking through the use of guided inquiry and example generation. | |||||||||||||||||||||||||||||||||||||
Learning Outcomes 1. interpret the formal mathematical definitions and statements which arise in analysis. 2. classify and describe the main components of the definitions or statements, and the motivation behind them. 3. give examples or counterexamples of important phenomena which are studied in mathematical analysis. 4. critique and explain the logical steps which are required to apply definitions or theorems to the phenomena which occur in mathematical analysis. 5. critique and explain the main logical arguments which occur in the proofs of a selection of theorems. 6. calculate important quantities which arise in mathematical analysis e.g. bounds of sets or sequences, convergence of sequences or series. | |||||||||||||||||||||||||||||||||||||
All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml |
|||||||||||||||||||||||||||||||||||||
Indicative Content and
Learning Activities Real numbersAxiomatic definition of the reals, Inequalities, modulus function, triangle inequality, bounded sets, supremum and infimumSequencesbounded sequences, monotone sequences, convergent sequences, Cauchy sequences, Convergence theoremsApplications of sequencesNewton approximation for square roots, Exponential function and the logarithm, (Iterated fractions)Application to Differential CalculusFormulation of sequential continuity, Derivative, and Riemann integral in terms of sequencesSeriesGeometric series, Telescoping series, Harmonic series, Leibnitz criterion, absolute convergence, Cauchy product, ratio test, root testPower seriesConvergence radius, Exponential function, Trigonometric functions, Differentiability, Integrability | |||||||||||||||||||||||||||||||||||||
| |||||||||||||||||||||||||||||||||||||
Indicative Reading List
| |||||||||||||||||||||||||||||||||||||
Other Resources None | |||||||||||||||||||||||||||||||||||||
Programme or List of Programmes | |||||||||||||||||||||||||||||||||||||
Archives: |
|