The notion of limit and convergence are two key ideas on which rest most of modern Analysis. This module aims to present these notions building on the experience gained by students in first year Calculus module. The context of our study is: limits and convergences of sequences and series of real numbers, and sequences and series of functions.
These classical results will be applied to derive properties of continues and differentiable functions. The module will introduce tools that are of importance in applications, for instance, power series expansions of functions, etc.
Prerequisites: (MATH1059 AND MATH1060) OR MATH1056
Aims and Objectives
Having successfully completed this module you will be able to:
- Prove using the definition that a given sequence converges to a given limit
- Determine whether a sequence of real numbers converges, either by evaluating the limit directly or by showing the sequence is bounded and monotone
- Find the Taylor series of a given function
- Determine a radius and interval of convergence for a given power series
- Understand when one can differentiate and integrate a power series
- Determine whether a given improper integral converges
- Define an improper integral
- Determine whether a series of positive terms converges, either by explicitly summing the series or by using a test, such as the comparison test, the ratio test, the root test, or the integral test
- Real numbers, bounded subsets of the reals, supremum and infimum, the Archimedean property of the real numbers.
- Sequences and their limits, arithmetic of limits, inequalities for limits.
- Bounded monotonic sequences, convergence and divergence criteria for sequences.
- The Bolzano-Weierstrass theorem. Completeness of the set of the reals.
- Continuity and uniform continuity of functions.
- Sequence and series of functions.
- Convergence of series by partial sums, geometric and harmonic series, algebra of series.
- Series with non-negative terms: comparison, integral, ratio, root tests.
- Absolute and conditional convergence, alternating test.
- Real power series, radius and interval of convergence; uniqueness, algebra, differentiation, integration of power series.
- Uniform convergence of sequences and series of continuous functions. The Weierstrass M-test. (if time)
- Cauchy mean value theorem, and convergence of Taylor series.
- Riemann Integral and the Fundamental Theorem of Calculus.
- Improper integrals.
Learning and Teaching
Teaching and learning methods
Lectures, problem classes, workshops, private study
|Total study time||150|
Resources & Reading list
NIKOLSKY S M. A Course of Mathematical Analsysis 1.
ALCOCK, L. How to think about analysis.
RUDIN W. Principle of Mathematical Analysis.
APPLEBAUM D. Limits, Limits Everywhere.
WADE R. An Introduction to Analysis.
HOWLAND J S. Basic Real Analysis.
JWA,JB,NW,BN,VP. Lecture notes.
This is how we’ll formally assess what you have learned in this module.
This is how we’ll assess you if you don’t meet the criteria to pass this module.
Repeat type: Internal & External