Aims & Learning Objectives:
Aims: To reinforce and extend the ideas and methodology (begun in the first year unit MA10004) of the analysis of the elementary theory of sequences and series of real numbers and to extend these ideas to sequences of functions.
By the end of the module, students should be able to read and understand statements expressing, with the use of quantifiers, convergence properties of sequences and series. They should also be capable of investigating particular examples to which the theorems can be applied and of understanding, and constructing for themselves, rigorous proofs within this context.
Suprema and Infima, Maxima and Minima. The Completeness Axiom.
Sequences. Limits of sequences in epsilon-N notation. Bounded sequences and monotone sequences. Cauchy sequences. Algebra-of-limits theorems.
Subsequences. Limit Superior and Limit Inferior. Bolzano-Weierstrass Theorem.
Sequences of partial sums of series. Convergence of series. Conditional and absolute convergence. Tests for convergence of series; ratio, comparison, alternating and n'th root tests.
Power series and radius of convergence.
Functions, Limits and Continuity. Continuity in terms of convergence of sequences. Algebra of limits. Brief discussion of convergence of sequences of functions.