MAS221 Analysis 2016-2017

Semester 2

Lecturer: Prof Sarah Whitehouse

This is the 2016-2017 course web page for semester 2 of MAS221 Analysis. It will be updated as the module progresses. For official course information, including timetabling, please consult the list of current modules.

Here is a link to the semester one webpage.

Semester 2 Course Information

A good grasp of Semester 1 material is essential to understanding the module in Semester 2. This semester we will continue the study of analysis of functions of one variable, building on what you saw in Semester 1. Later on we will generalise to the study of functions of several variables.

Here is a summary of the material we will cover in semester 2, with a rough indication of how many lectures will be devoted to each topic.

  • Convergence of series in terms of the sequence of partial sums.  Examples, including geometric series and harmonic series.  Tests for convergence: comparison test, absolute convergence, ratio test. (3 lectures)
  • Definition of the Riemann integral. Basic properties. Examples. Continuous functions are Riemann integrable. The fundamental theorem of calculus. Improper integrals. (3 lectures)
  • Sequences and series of functions; pointwise and uniform convergence.  Proof that a uniform limit of a sequence of continuous functions is continuous.  Example to show the same is not true of pointwise limits. Uniform continuity. The uniform convergence theorem for integrals, and application to differentiation.  Series of functions and the Weierstrass $M$-test for uniform convergence.   Example of an everywhere continuous but nowhere differentiable function. (5 lectures)
  • Applications I; power series, radius of convergence, termwise differentiation. Exponential function and e. (2 lectures)
  • Sequences in R^k.  Distances in R^k. Convergence of sequences in R^k. Open balls and open sets.  Intersection and unions.  Examples. Cauchy sequences revisited. (3 lectures)
  • Functions of several variables. Images and pre-images, continuity,  algebra of continuous functions. Examples, including polynomials. Uniform continuity. Examples. (3 lectures)
  • Applications II.  Differentiation under the integral sign. Applications to the B and Gamma functions. (2 lectures)


There was an in-class test in the Thursday lecture of week 1.

Important comment: The test questions are comparable to the "easy marks" part of the exam; you need to be getting pretty much all of these marks to pass the exam.

Lecture notes

Here are the lecture notes for the whole of the semester. (Updated version 8 May 2017 -  a couple of typos corrected since version of 3 April.)

This version contains the statements of definitions and results, but not proofs or solutions to examples. You are expected to attend the lectures and take your own notes of the proofs and the solutions to examples. (I expect to supply some version of this material in typed form eventually, but this is unlikely to contain everything that I explain in the lectures. In particular, pictures and graphs are important to aid understanding of the material. I expect to draw lots of these in lectures, but not to produce a printed version of them.)

Here is a fuller version of the lecture notes, with solutions to examples and proofs.  


Here is the complete set of exercises for the semester. (Version of 3 April 2017 - some typos corrected.)

Problem classes

These take place on Tuesdays (at 11 or 2) in weeks 2, 4, 6, 8, 10 and 12.

Week 2 class: we'll focus on the following problems: Chapter 1, Qs 1, 3, 4, 5, 6.
Week 4 class: we will go through Q2(b) from the test and then focus on Chapter 2, Qs 5, 3 4(a), (b), (d), 6.
k 6 class: Chapter 3, Qs 3, 4 and 6.
Week 8 class: Chapter 3, Qs 11, 13. Chapter 4, Q1.
Week 10 class: Chapter 5, Q2, 5, 6, 8 (9, 11 if time).
Week 12 class: Chapter 6, Q3, 4, 5, 7.

Homework schedule

Details of questions set and hand-in arrangements will appear here as the module progresses.

Homework 5 from semester 1 was Qs 83, 84, 88, 91 and 98. This was due in at the week 2 tutorials.
Homework 1 for this semester was Chapter 1, Qs 7, 8 and 9 and Chapter 2, Question 2. This was due in at the week 4 tutorials.
Homework 2 for this semester is Chapter 2, Qs 7 and 12 and Chapter 3, Q2.
This is due in at the week 6 tutorials.
Homework 3:
 Chapter 3, Q5(a),(b), Q7 and Q10. This is due in at the week 8 tutorials.
Homework 4:  Chapter 4, Qs 2,3, Chapter 5, Q1.
 This is due in at the week 10 tutorials.

No further homework for marking, but I suggest looking at Chapter 5 Q7, Chapter 6, Q1 and Q2 before the week 12 tutorials.


Solutions to Chapter 0 Problems: Revision
Solutions to Chapter 1 Problems: Series
Solutions to Chapter 2 Problems: Riemann Integration
Solutions to Chapter 3 Problems: Sequences and series of functions
Solutions to Chapter 4 Problems:  Applications I
Solutions to Chapter 5 Problems: Sequences in R^k
Solutions to Chapter 6 and 7 Problems: Functions of several variables and Applications II


The last two years' exam papers are available from here.  Solutions to these papers are below.

MAS221 exam solutions 2014-15

MAS221 exam solutions 2015-16

Further exam information is here.

Contact Details

Prof Sarah Whitehouse
Room J6c, Hicks Building
Telephone: 0114 2223870  (internal extension: 23870)

Office Hour:  Monday 2-3 (but not week 4) or by appointment

You can always make an appointment by email if you cannot come to an office hour.

Last updated: 12 May 2016