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MA2509: ANALYSIS II (2017-2018)

Last modified: 23 Aug 2017 15:31


Course Overview

Analysis provides the rigourous, foundational underpinnings of calculus. This course builds on the foundations in Analysis I, and explores the notions of Riemann integrability, Cauchy sequences, sequences of functions, and power series.

The techniques of careful rigourous argument seen in Analysis I will be further developed. Such techniques will be applied to solve problems that would otherwise be inaccessible. As in Analysis I, the emphasis of this course is on valid mathematical proofs and correct reasoning.

Course Details

Study Type Undergraduate Level 2
Session Second Sub Session Credit Points 15 credits (7.5 ECTS credits)
Campus Old Aberdeen Sustained Study No
Co-ordinators
  • Dr Alexey Sevastyanov

Qualification Prerequisites

  • Programme Level 2

What courses & programmes must have been taken before this course?

What other courses must be taken with this course?

None.

What courses cannot be taken with this course?

None.

Are there a limited number of places available?

No

Course Description

- Riemann integrability: Riemann sums, basic properties, the Fundamental Theorem of Calculus, improper integrals
- Cauchy sequences: Cauchy's characterisation of convergent sequences, Cauchy criterion for series, rearrangements of series
- Sequences of functions: pointwise convergence, uniform convergence, properties of limits of functions, Dini's Theorem, series of functions
- Power series: convergence, continuity, differentiability, integrability, Taylor series, manipulations of power series

 

Syllabus

  • Riemann integrability and the Riemann integrals.
  • Integrability of continuous functions; characterisations of  integrability; properties of the integral.
  • Cauchy sequences, upper and lower limits, applications to infinite  series.
  • Cauchy criterion; limsup and  liminf and their key properties; the equivalence of absolute convergence and nonconditional convergence.
  • Sequences and series of functions:
  • Pointwise and uniform convergence; examples of pointwise converging sequences with bad  behaviour (regarding continuity, differentiation, integration);
  • theorems about good behaviour under uniform convergence; Weierstrass' M-test;
  • Dini's Theorem; Dominated Convergence Theorem; Pointwise limits of continuous functions (involving the Baire Category Theorem).
  • Taylor series. Computing radius of convergence using the limsup root test; uniform convergence of power series; Lagrange's form of the remainder; products of power series.

Further Information & Notes

Course Aims
To further develop understanding of the concepts, techniques, and tools of calculus. Calculus is the mathematical study of variation. This course emphasises integral calculus, sequences and series, and introduces multivariable calculus. Applications to the theory of functions will be discussed.
 
Learning Objectives
By the end of this course the student should:
  • be able to state the main definitions and theorems of the course;
  • be able to prove most results from the course;
  • understand Riemann integration and theorems about the Riemann integral;
  • be able to apply techniques for showing integrability or non-integrability of functions;
  • understand Cauchy sequences and their relationships to convergent sums and the Cauchy criteria for convergence of series;
  • be able to distinguish between pointwise and uniform convergence of sequences of functions;
  • be able to compute Taylor series, compute the interval of convergence of power series, and use Taylor's theorem to estimate functions by polynomials.

Degree Programmes for which this Course is Prescribed

  • BSc Applied Mathematics
  • BSc Computing Science-Mathematics
  • BSc Mathematics
  • BSc Mathematics with French
  • BSc Mathematics with Gaelic
  • MA Business Management - Mathematics
  • MA Mathematics
  • MA Mathematics with Gaelic
  • Mathematics Joint
  • Mathematics Major
  • Mathematics Minor

Contact Teaching Time

44 hours

This is the total time spent in lectures, tutorials and other class teaching.

Teaching Breakdown


Assessment

 

1st attempt - 1 two-hour written examination (80%); in-course assessment (20%).

Resit – 1 two-hour written examination paper. Maximum of written exam (100%) or written exam (80%) with carried forward in-course assessment (20%).

Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

Feedback

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact Course Coordinator for feedback on the final examination.

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