Mathematical Sciences

MATH2039 Analysis

Module Overview

Core for MMath, BSc Mathematics.

Compulsory for  BSc Financial Mathematics, BSc Mathematical Studies, Mathematics with Actuarial Science, Mathematics with Economics, Mathematics with Finance, Mathematics with Management Sciences,  Mathematics with Music, Mathematics with Operational Research, Mathematics with Statistics, Mathematics and Modern Language.

Module Details

Title: Analysis
Code: MATH2039
Year: 2
Semester: 1

CATS points: 15 ECTS points: 7.5
Level: Undergraduate
Co-ordinator(s):

Pre-requisites and / or co-requisites

ModuleCodeYear
Calculus IMATH10501
Calculus IIMATH10511
Differential EquationsMATH10521

Aims and objectives

  • Determine whether a sequence of real numbers converges, either by evaluating the limit directly or by showing the sequence is bounded and monotone;
  • prove using the definition that a given sequence converges to a given limit;
  • 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;
  • Calculate radius of convergence for complex power series;
  • Understand when one can differentiate and integrate power series;
  • Apply the Weierstrass M-test;
  • Find the Taylor series of a given function and evaluate the error;
  • Determine if the Taylor series of a function converges;
  • Approximate continuous functions by piecewise linear functions;
  • Determine whether a given improper integral converges.

Syllabus

  • 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. Upper and lower limits.
  • The Weierstrass theorem. Completeness of the set of the reals.
  • [If time] Extension of elementary functions from the rationals to the real numbers.
  • Sequences and continuity of functions.
  • Maximum properties of continuous functions, uniform continuity.
  • Convergence of series by partial sums, geometric and harmonic series, algebra of series, Euler’s constant.
  • Series with non-negative terms: comparison, integral, ratio, root tests.
  • Absolute and conditional convergence, alternating test.
  •  Complex series, Riemann zeta function
  • Real and Complex power series, radius and interval of convergence; uniqueness, algebra, differentiation, integration of power series.
  • Irrationality of e, transcendental numbers
  • Infinite products.
  • Uniform convergence of sequences and series of continuous functions. The Weierstrass M-test.
  • Cauchy mean value theorem, and convergence of Taylor series.
  • Approximating continuous functions by piecewise linear functions. Construction of the definite integral. (If time). (We might include the Fundamental Theorem of Calculus here.)
  • Improper integrals, differentiation of integrals.
  • Gamma function, Stirling’s approximation, Beta function. 
  • (If time) Divergent series, asymptotics.

Learning and teaching

Study time allocation

Contact hours: 4
Private study hours: 4
Total study time: 8 hours

Teaching and learning methods

Lectures, problem classes, private study

Resources and reading list

ADAMS, Calculus 

JWA,JB,NW, BN, Lecture notes

WADE, An Introduction to Analysis

Assessment

Assessment methods

  • 80% Written Examination;
    • PURPOSE OF ASSESSMENT: To allow students to demonstrate that they have understood the material in the module and can carry out basic calculations correctly
    • FEEDBACK: Generic feedback on the performance of the class as a whole will be provided to students by e-mail
  • 10% Class Tests;
    • PURPOSE OF ASSESSMENT: To provide students with a snapshot of their progress and provide feedback on where they should direct their study efforts
    • FEEDBACK: Students will be provided with model solutions and an opportunity to look at their marked tests and ask questions
  • 10% Coursework;
    • PURPOSE OF ASSESSMENT: To provide regular practice, supported by feedback, on the basic techniques taught in the module. Larger pieces of coursework provide students with the opportunity to demonstrate higher level understanding of the module material
    • FEEDBACK: Students will be provided with model solutions and marked copies of the work with comments made by the marker. They will also have opportunities to ask questions about the work in tutorial sessions
  • Referral arrangements: Written Examination .