MMath Mathematics and Statistics / Course details

The notion of limit underlies the differential and integral calculus, a central topic in Mathematics. A good understanding of this concept was developed in the early nineteenth century, many years after the calculus was first used, and this is essential for more advanced calculus. The main purpose of this course is to provide a formal introduction to the concept of limit in its simplest setting: sequences and series.

The aims of this course are to develop an understanding of convergence in its simplest setting. To explain the difference between a sequence and a series in the mathematical context. To lay foundations for further investigation of infinite processes, in particular differential and integral calculus.

On successful completion of this module students will be able to:

• express correctly the definitions of the basic concepts from the course unit, for example the definition of the limit of a sequence,
• write short simple proofs involving the basic concepts from the course unit and apply the Completeness property of the Reals where needed,
• decide on the correctness or otherwise of statements involving the basic concepts from the course unit, providing justifications or counterexamples as appropriate,
• find the limit of a wide class of sequences of real numbers,
• decide on convergence or divergence of a wide class of series of real numbers or power series with real coefficients,
• find the radius and interval of convergence of a power series.

1.Convergent sequences, properties of the class of convergent sequences, including Algebra of Limits. Sequences diverging to infinity, the Reciprocal Rule, subsequences and the subsequence strategy. Ratio Test, L'Hopital's Rule. The Monotone Convergence Theorem.

2.Convergent series, the geometric series and the harmonic series. Series with non-negative terms, the Comparison Test, the Ratio Test and the Integral Test. The Alternating Series Test, absolute and conditional convergence of series, power series and radius of convergence.

• Coursework; Weighting within unit 20%
• Examination; Weighting within unit 80%

Feedback supervision will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

L. Alcock, How to Think about Analysis, Oxford University Press, 2014. ISBN-13:978-0198723530

 R. Haggerty, Fundamentals of Mathematical Analysis, Addison Wesley, 1993

 V. Bryant. Yet Another Introduction to Analysis, C.U.P, 1990.

This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.  

Please see Blackboard / course unit related emails for any further updates