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### 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 Level Undergraduate 2 Second Sub Session 15 credits (7.5 ECTS credits) Old Aberdeen No Dr Alexey Sevastyanov

### Qualification Prerequisites

• Programme Level 2

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

• MA2009 Analysis I (Studied)

None.

None.

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.

### 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.

### 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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