M2: Analysis I - Sequences and Series (2021-22)
Section outline
-
-
-
-
This block of work corresponds to Weeks 1 and 2 (the equivalent of 3 lectures), and is designed for tutorials in Week 3. It's about the real numbers, arithmetic, ordering, and modulus.
The relevant sections of the lecture notes are Sections 2, 3, 4, 5, 6, 7.
For this block of work, from the lecture videos please watch the introductory video 1 (length 14:42), and then watch videos 2 (length 14:51), 3 (length 37:13), 4 (length 14:46), 5 (length 25:02), 6 (length 18:04), 7 (length 3:50).
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? .
-
This block of work corresponds to Week 3 (the equivalent of 2 lectures), and is designed for tutorials in Week 4. It's about the modulus, supremum, infimum, and countability.
The relevant sections of the lecture notes are Sections 8, 10, 11, 12, 13, 14. (Note that there is no section 9.)
For this block of work, from the lecture videos please watch video 8 (length 4:36) and try the quiz (Section/Video 8) Upper and lower bounds, then watch videos 10 (length 29:31), 11 (length 21:19), 12 (length 15:45), 13 (length 21:34), 14 (length 21:45). Note that there is no video 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 optional activities for this block of work: (Sheet 2) Starter - sup and max, and (Sheet 2) Pudding - Countability.
-
This block of work corresponds to Week 4 (the equivalent of 2 lectures), and is designed for tutorials in Week 5. It's about convergence, infinite limits, and complex sequences.
The relevant sections of the lecture notes are Sections 15, 16, 17, 18, 19, 20.
For this block of work, from the lecture videos please watch videos 15 (length 16:57), 16 (length 43:54), 17 (length 26:59), 18 (length 20:41), 19 (length 26:26), 20 (length 5:35).
In this block, I took advantage of the video format to include some additional worked examples that I wouldn't have had time to go through in face-to-face lectures. If you're pushed for time, then you could skip through these, but I thought that some of you might find them useful.
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.
In Video 18, I mentioned a couple of useful resources. One was 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 was 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 (the equivalent of 2 lectures), and is designed for tutorials in Week 6. It's about subsequences, the algebra of limits, and monotonic sequences.
The relevant sections of the lecture notes are Sections 21, 22, 23, 24, 25.
For this block of work, from the lecture videos please watch videos 21 (length 14:22), 22 (length 21:02), 23 (length 21:35), 24 (length 10:07), then please try the quiz (Section/Video 24) Big O and little o notation, then please watch video 25 (length 30:56).
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 two optional activities for this block of work: (Sheet 4) Starter - Sequences and subsequences and (Sheet 4) Pudding - Limit of limits.
-
This block of work corresponds to Week 6 (the equivalent of 2 lectures), 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.
The relevant sections of the lecture notes are Sections 26, 27, 28, 29, 30.
For this block of work, from the lecture videos please watch videos 26 (length 14:33), 27 (length 18:14), 28 (length 21:00), 29 (length 27:26), 30 (length 31:42).
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 - Bolzano-Weierstrass Theorem and (Sheet 5) Pudding - Some sequences and series .
-
This block of work corresponds to Week 7 (the equivalent of 2 lectures), and is designed for tutorials in Week 8. It's about convergence of series and tests for convergence.
The relevant sections of the lecture notes are Sections 31, 32, 33, 34.
For this block of work, from the lecture videos please watch videos 31 (length 14:45), 32 (length 28:20), 33 (length 22:43), 34 (length 14:58).
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 (the equivalent of 2 lectures), and is designed for tutorials at the start of Hilary Term. It's about conditionally convergent series and power series.
The relevant sections of the lecture notes are Sections 34, 35, 36, 37.
For this block of work, from the lecture videos please watch videos 34 (length 14:58), 35 (length 19:42), 36 (length 27:38), 37 (length 22:33). There is also an optional video 38 (length 17:43), with tips on what you might usefully do regarding Analysis I over the vacation.
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.
-