|Credits: 10||Convenor: Dr C. Eaton||Semester: 1 (weeks 1-6)|
|Assessment:||Individual and group coursework: 20%||One and a half hour hour exam: 80%|
The key to rigourising our understanding of such functions is through the study of infinite series: these are the fundamental objects studied in the course. Now an infinite series (a sum of infinitely many numbers) is a questionable object--we can only sum things together one at a time, so how can we hope to make anything meaningful from the sum of infinitely many quantities. This module explains how some infinite series can be considered to represent bona fide quantities--such series are called convergent, and how such series can be distinguished from those that are divergent. It then becomes possible to define functions, in particular the trigonometric and exponential functions, as infinite series. And having done so, it is natural to want to differentiate and integrate such functions. But (and haven't we heard this before) what is really meant by differentiation and integration? The module also puts these topics on a firm footing with rigorous definitions based on the notions of a limit (remember MC146) and of, guess what, infinite series.
To understand, reconstruct and apply the main results and proofs covered in the module.
To know the definition of convergence for infinite series, and test for convergence using standard tests.
To know the formal definitions of differentiation and Riemann integration.
Subsequences, Bolzano-Weierstrass Theorem, Cauchy sequences.
Infinite series, geometric series, harmonic series, comparison test, ratio test, alternating series, conditional convergence, sums and products of series, power series, radius of convergence.
Local maximum and minimum, Rolle's theorem, Mean Value Theorem, differentiation of power series, Taylor's series and theorem.
Dissection of an interval, upper and lower sums, refinement, upper and lower integrals, integrable functions, fundamental theorem of calculus, limitations of the theorem of Riemann integration.
R. Haggarty, Fundamentals of Mathematical Analysis, Adison Wesley.
M. Spivak, Calculus, Benjamin-Cummings.
D. Stirling, Mathematical Analysis: a fundamental and straightforward approach, Ellis Horwood.