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Sequences and Series


Unit code: MATH10242
Credit Rating: 10
Unit level: Level 1
Teaching period(s): Semester 2
Offered by School of Mathematics
Available as a free choice unit?: N

Requisites

Prerequisite

Aims

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.

Overview

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.

Learning outcomes

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.

 

 

Assessment methods

  • Other - 20%
  • Written exam - 80%

Assessment Further Information

  • Coursework; Weighting within unit 20%
  • Two hours end of semester examination; Weighting within unit 80%

Syllabus

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.

Recommended reading

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

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

Feedback methods

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.

Study hours

  • Lectures - 22 hours
  • Tutorials - 11 hours
  • Independent study hours - 67 hours

Teaching staff

Mike Prest - Unit coordinator

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