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The University of Southampton

MATH2039 Analysis

Module Overview

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.

Aims and Objectives

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

  • 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
  • Define an improper integral
  • Determine whether a given improper integral converges
  • Determine a radius and interval of convergence for a given power series
  • Understand when one can differentiate and integrate a power series
  • Find the Taylor series of a given function


• 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

Independent Study96
Total study time150

Resources & Reading list

ALCOCK, L. How to think about analysis. 

NIKOLSKY S M. A Course of Mathematical Analsysis 1. 

WADE R. An Introduction to Analysis. 

ADAMS. Calculus. 

HOWLAND J S. Basic Real Analysis. 

JWA,JB,NW,BN,VP. Lecture notes. 

RUDIN W. Principle of Mathematical Analysis. 

APPLEBAUM D. Limits, Limits Everywhere. 



MethodPercentage contribution
Coursework 40%
Written assessment 60%


MethodPercentage contribution
Written assessment 100%

Repeat Information

Repeat type: Internal & External

Linked modules

Prerequisites: (MATH1059 AND MATH1060) OR MATH1056


Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at

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