In these lectures we study the real and complex numbers, and study their properties, particularly completeness; define and study limits of sequences, convergence of series, and power series.
See the examinable syllabus.
Prof. Frances Kirwan
Students will have:
(i) an ability to work within an axiomatic framework;
(ii) a detailed understanding of how Cauchy's criterion for the convergence of real and complex sequences and series follows from the completeness axiom for \(\mathbb{R}\), and the ability to explain the steps in standard mathematical notation;
(iii) knowledge of some simple techniques for testing the convergence of sequences and series, and confidence in applying them;
(iv) familiarity with a variety of well-known sequences and series, with a developing intuition about the behaviour of new ones;
(v) an understanding of how the elementary functions can be defined by power series, with an ability to deduce some of their easier properties.
Real numbers: arithmetic, ordering, suprema, infima; the real numbers as a complete ordered field. Definition of a countable set. The countability of the rational numbers. The reals are uncountable. The complex number system. The triangle inequality.
Sequences of real or complex numbers. Definition of a limit of a sequence of numbers. Limits and inequalities. The algebra of limits. Order notation: \(O\), \(o\).
Subsequences; a proof that every subsequence of a convergent sequence converges to the same limit; bounded monotone sequences converge. Bolzano--Weierstrass Theorem. Cauchy's convergence criterion.
Series of real or complex numbers. Convergence of series. Simple examples to include geometric progressions and some power series. Absolute convergence, Comparison Test, Ratio Test, Integral Test. Alternating Series Test.
Power series, radius of convergence. Examples to include definition of and relationships between exponential, trigonometric functions and hyperbolic functions.