MATH1051
Lecturer : Norbert Peyerimhoff
Term : Epiphany 2015
Literature
The following is a list of books on which the course is based. Although we will not follow a book strictly, most of the material can be found in them and they may sometimes offer a different approach to the material.Resources
Tutorial and Homework problems
| Week | Tutorial Problems | Extra Tutorial Problems | Homework Problems | Problems Class | |
| Week 11 | No tutorials | No tutorials | 104, 110, 111, 114 | No Problems Class | |
| Week 12 | 102, 113, 116, 117 | 101, 105, 108, 112 | 115, 120, 124 | No Problems Class | |
| Week 13 | No tutorials | No tutorials | 130abfh, 134, 136 | 112, 144, 145 | |
| Week 14 | 128, 135, 137 | 126, 127, 129, 130 | 138bce, 141, 142, 150bd | No Problem Class | |
| Week 15 | No tutorials | No tutorials | 152, 154, 156 | 155, 157 | |
| Week 16 | 160, 167 | 168, 158 | 159, 161, 162, 163 | No Problem Class | |
| Week 17 | No tutorials | No tutorials | 165, 169ac, 175bde | 166, 175ac, 178, 180 | |
| Week 18 | 164, 182, 185 | 183, 189 | 181, 184, 186, 188 | No Problem Class | |
| Week 19 | No tutorials | No tutorials | 187, 192adef, 195, 204 | 192bc, 203, 208bc, 210 |
Content of Lectures
| Date | Content |
| Monday, 12 January 2015 (Week 11) | Collections |
| Tuesday, 13 January 2015 (Week 11) | Definition of differentiability, examples, basic properties |
| Friday, 16 January 2015 (Week 11) | Rolle, Mean Value Theorem, Generalised Mean Value Theorem, L'Hopital's Rule |
| Monday, 19 January 2015 (Week 12) | Discussion of Collections |
| Tuesday, 20 January 2015 (Week 12) | Definition of an (infinite) series, geometric and harmonic series as examples, start with convergence considerations |
| Friday, 23 January 2015 (Week 12) | Examples for Comparison Test, proof of Comparison Test, absolute and conditional convergence, Absolute Convergence Theorem |
| Monday, 26 January 2015 (Week 13) | Examples for Absolute Convergence Theorem, Alternating Sign Test, Integral Test and Ratio Test, examples |
| Tuesday, 27 January 2015 (Week 13) | Examples for the Ratio Test, n-th Root Test, Rearrangements and Riemann Rearrangement Theorem about conditionally convergent series |
| Friday, 30 January 2015 (Week 13) | Problems Class |
| Monday, 2 February 2015 (Week 14) | Rearrangements of absolutely convergent series, Cauchy Product Theorem, Complex Series |
| Tuesday, 3 February 2015 (Week 14) | Examples (e.g., Euler's Identity), partitions, lower and upper Riemann sums |
| Friday, 6 February 2015 (Week 14) | Examples of lower and upper Riemann sums, Riemann sums of refinements of partitions, Riemann integral, example of a not integrable function, criterion for Riemann integrability |
| Monday, 9 February 2015 (Week 15) | Proof of criterion for Riemann integrability, example, Riemann integrability of monotone and continuous functions |
| Tuesday, 10 February 2015 (Week 15) | Short discussion of uniform continuity, properties of the Riemann integral |
| Friday, 12 February 2015 (Week 15) | Problems Class |
| Monday, 16 February 2015 (Week 16) | Remaining part of proof of properties of the Riemann integral, Definition of C(I) and C^k(I), Fundamental Theorem of Calculus, Mean Value Theorem for Integrals |
| Tuesday, 17 February 2015 (Week 16) | Proof of Mean Value Theorem for Integrals, complex integrals, introduction to improper integrals |
| Friday, 20 February 2015 (Week 16) | Examples of improper integrals, comparison test for improper integrals, examples, definition of absolute convergence of improper integrals |
| Monday, 23 February 2015 (Week 17) | Absolute Convergence Theorem with proof, examples, example f_n(x)=x^n on [0,1] for pointwise convergence |
| Tuesday, 24 February 2015 (Week 17) | Definition of pointwise convergence of sequence of functions and pointwise limit, idea and definition of uniform convergence of sequence of functions, example, uniform convergence implies pointwise convergence, uniform convergence preserves continuity with proof |
| Friday, 27 February 2015 (Week 17) | Problems Class |
| Monday, 2 March 2015 (Week 18) | Properties of uniform convergence (integrals of uniformly convergent functions and C^1-functions whose derivatives are uniformly convergent), example |
| Tuesday, 3 March 2015 (Week 18) | Complex power series, radius of convergence, examples, methods to calculate radius of convergence, based on ratio test and n-th rott test, examples, first results on absolute convergence of power series |
| Friday, 6 March 2015 (Week 18) | Existence of radius of convergence for any power series, Weierstrass M-Test for proof of uniform convergence, proof of uniform convergence of real power series on compact intervals [-r,r] with r < radius of convergence |
| Monday, 9 March 2015 (Week 19) | Proof of differentiability of real power series and that derivative agrees with termwise differentiated power series, Identity Theorem for real power series and proof, Definition of Taylor series |
| Tuesday, 10 March 2015 (Week 19) | Examples of Taylor series, Lagrange Form of Remainder Term to decide where Taylor series represents the original function |
| Friday, 13 March 2015 (Week 19) | Problems Class |