Sequences and Series
|Unit level:||Level 1|
|Teaching period(s):||Semester 2|
|Offered by||School of Mathematics|
|Available as a free choice unit?:||N
- MATH10101 - Foundations of Pure Mathematics A (Compulsory)
- MATH10111 - Foundations of Pure Mathematics B (Compulsory)
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.
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.
On successful completion of this module students will be able to
know the definition of the limit of a sequence.
be able to find the limit of a wide class of sequences.
be able to decide on convergence or divergence of a wide class of series.
know that a power series has a radius of convergence, and know how to find it.
Future topics requiring this course unit
Second year courses: Real and Complex Analysis, Applied Analysis courses, Numerical Analysis courses.
- Other - 20%
- Written exam - 80%
Assessment Further Information
- Coursework; Weighting within unit 20%
- Two hours end of semester examination; Weighting within unit 80%
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.
R. Haggerty, Fundamentals of Mathematical Analysis, Addison Wesley, 1993
V. Bryant. Yet Another Introduction to Analysis, C.U.P, 1990.
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.
- Lectures - 22 hours
- Tutorials - 11 hours
- Independent study hours - 67 hours