M2: Analysis I  Sequences and Series (202223)
Topic outline


This block of work corresponds to Weeks 1 and 2 (3 lectures), and is designed for tutorials in Week 3. It's about the real numbers, arithmetic, ordering, and modulus. Lecture notes Sections 2, 3, 4, 5, 6, 7.
The problems sheet is below as a pdf attachment, and is also available as a Sheet 1 Moodle version for those who find that more convenient/accessible.
There are also optional activities for this block of work: a quiz (Sheet 1) Starter 1  Axioms for R and (Sheet 1) Starter 2  Name that axiom and (Sheet 1) Pudding  Field or Not a Field? . There's also an extra optional quiz Highlights of the Greek alphabet. It's not directly related to this block of work as we shan't use Greek letters till slightly later in term, but it's available to you at any time and might be useful if you don't yet feel confident with naming and writing Greek letters.

This block of work corresponds to Week 3, and is designed for tutorials in Week 4. It's about the modulus, supremum, infimum, and countability. Lecture notes Sections 8, 10, 11, 12, 13, 14 (note that there is no section 9).
The problems sheet is below as a pdf attachment, and is also available as a Sheet 2 Moodle version for those who find that more convenient/accessible.
There are also three optional activities for this block of work: (Sheet 2) Starter  Upper and lower bounds, (Sheet 2) Starter  sup and max, and (Sheet 2) Pudding  Countability.

This block of work corresponds to Week 4, and is designed for tutorials in Week 5. It's about convergence, infinite limits, and complex sequences. Lecture notes Sections 15, 16, 17, 18, 19, 20.
The problems sheet is below as a pdf attachment, and is also available as a Sheet 3 Moodle version for those who find that more convenient/accessible.
Here are a couple of resources you might find useful. One is self explanation training  if you head to that link you'll find a pdf booklet for students, which might take you about half an hour to work through. The other is How to think about Analysis, by Lara Alcock. This is on the reading list for this course, where you can click through to an electronic version available via the university library.
There is also an optional activity for this block of work: (Sheet 3) Pudding  Mandelbrot set.

This block of work corresponds to Week 5, and is designed for tutorials in Week 6. It's about subsequences, the algebra of limits, and monotonic sequences. Lecture notes Sections 21, 22, 23, 24, 25.
The problems sheet is below as a pdf attachment, and is also available as a Sheet 4 Moodle version for those who find that more convenient/accessible.
There are also three optional activities for this block of work: (Sheet 4) Starter  Sequences and subsequences, (Sheet 4) Starter  Big O and little o notation, and (Sheet 4) Pudding  Limit of limits.

This block of work corresponds to Week 6, and is designed for tutorials in Week 7. It's about Cauchy sequences, series (convergence and absolute convergence), properties of e, and the Alternating Series Test. Lecture notes Sections 26, 27, 28, 29, 30.
The problems sheet is below as a pdf attachment, and is also available as a Sheet 5 Moodle version for those who find that more convenient/accessible.
There are also two optional activities for this block of work: (Sheet 5) Starter  BolzanoWeierstrass Theorem and (Sheet 5) Pudding  Some sequences and series .

This block of work corresponds to Week 7, and is designed for tutorials in Week 8. It's about convergence of series and tests for convergence. Lecture notes Sections 31, 32, 33, 34.
The problems sheet is below as a pdf attachment, and is also available as a Sheet 6 Moodle version for those who find that more convenient/accessible.

This block of work corresponds to Week 8, and is designed for tutorials at the start of Hilary Term. It's about conditionally convergent series and power series. Lecture notes Sections 34, 35, 36, 37.
The problems sheet is below as a pdf attachment, and is also available as a Sheet 7 Moodle version for those who find that more convenient/accessible.
There is also one optional activity for this block of work: (Sheet 7) Pudding  Stirling's formula.


This quiz is designed to help you learn the letters from the Greek alphabet that you are most likely to need as a mathematician. You can tackle it at any stage of the term.
