Webpage of the Course 400297 T1 2023/24
Welcome to the Course of Numbers, Sequences and Series 400297 for the BSc in Mathematics at the University of Hull, academic year 2023/24. This is an introductory course of Mathematical Analysis. We will investigate the properties of real numbers and explore the concept of limit.
If you have any questions please feel free to email me. We will use the Tutorials to address homework and exam preparation. In addition, please do not hesitate to attend office hours.
All the course information will be posted on this page, as well as on Canvas. The links to the reference material are:
We have two Lectures and one Tutorial per week. The lectures are 2 hours long, and the tutorial is 1 hour long.
Lecture 1: Thursdays 9:00-11:00 with venues as follows
Lecture 2: Thursdays 16:00-18:00 in Wilberforce Building - Lecture Theatre 12
Tutorial: Fridays 9:00-10:00 in Brynmor Jones Library - Teaching Room 1
This course will be assessed as follows:
Each Homework paper is worth 100 points. The final Homework grade will be computed averaging the best 4.
Homework papers will be posted on Canvas.
Homework papers must be submitted on Canvas by 14:00 on the Due Date. The due dates are:
Homework papers submitted outside of Canvas or after the deadline will NOT BE MARKED
Numbers: Number sets, algebraic operations and fields. Axiomatic construction of the real numbers. Completeness of \(\mathbb{R}\). Countability of \(\mathbb{Q}\). Complex numbers: definition and main properties, complex plane, solving polynomial equations.
Sequences: Definition of convergence. Algebra of limits. Limit tests. Complex sequences.
Series: Known series. Convergence of series. Nonnegative series. Testing convergence of general series. Rearrangements.
Lecture Notes: Available here
Main Book: Bartle and Sherbert
Other resources: Abbott
The course Lecture Notes are available here. Each week we have 2 Lectures of 2h each and 1 Tutorial of 1h, for a total of 5h of contact.
| Lesson | Date | Time | Topics |
|---|---|---|---|
| 1 | 28/09/23 | 09:00 - 11:00 | Intro: Canvas page, Lecture Notes, Assignments, Assessment. What is $\mathbb{R}$? Numbers $\mathbb{N}, \mathbb{Z}, \mathbb{Q}$. Proof that $\sqrt{2} \notin \mathbb{Q}$. |
| 2 | 28/09/23 | 16:00 - 18:00 | Preliminaries Sets, Basic Logic, Operations on Sets, Binary relations, Equivalence relation with examples. |
| 3 | 29/09/23 | 09:00 - 10:00 | Partial Order relation. Total order. Intervals. Functions. |
| 4 | 05/10/23 | 09:00 - 11:00 | Absolute value. Geometric meaning. Basic lemma on absolute value. Triangle inequality with proof. Proofs in Mathematics. Example of proof involving $\varepsilon$. |
| 5 | 05/10/23 | 16:00 - 18:00 | Induction. Recurrence sequence. Correction of exercises in Tutorial 1. |
| 6 | 06/10/23 | 09:00 - 10:00 | Binary operations. Fields. Field with $2$ elements. Uniqueness of neutral element and inverse. $\mathbb{N}$ and $\mathbb{Z}$ are not fields. $\mathbb{Q}$ is a field. |
| 7 | 12/10/23 | 09:00 - 11:00 | Ordered Fields. $\mathbb{Q}$ is ordered field. Partition of a set, Cut of a set, Cut Property. Proof that $\mathbb{Q}$ does not have the Cut Property. |
| 8 | 12/10/23 | 16:00 - 18:00 | Upper bound, bounded above, sup. Uniqueness of sup. Max. Lower bound, inf, min. Relation between inf and sup. Proof that $\mathbb{Q}$ is not complete. Axiom of Completeness. |
| 9 | 13/10/23 | 09:00 - 10:00 | Correction of exercises in Tutorial 2. |
| 10 | 19/10/23 | 09:00 - 11:00 | Cut property is equivalent to Completeness. $\mathbb{R}$ as ordered complete field. Archimedean property: 2 Versions. Nested Interval Property. |
| 11 | 19/10/23 | 16:00 - 18:00 | Equivalent formulation of sup and inf. Sup, inf, max, min of interval $(a,b)$. Correction of Homework 1. |
| 12 | 20/10/23 | 09:00 - 10:00 | Correction of exercises in Tutorial 3. |
| 13 | 26/10/23 | 09:00 - 11:00 | Examples of calculation of sup and inf. $\mathbb{N}$ as inductive subset of $\mathbb{R}$. Properties. Induction. Definition of $\mathbb{Z}$, $\mathbb{Q}$ and properties. |
| 14 | 26/10/23 | 16:00 - 18:00 | Density of $\mathbb{Q}$ and irrationals in $\mathbb{R}$. Existence of $k$-th roots. Bijective functions. Examples. Cardinality. Subsets of countable sets. |
| 15 | 27/10/23 | 09:00 - 10:00 | Correction of Tutorial 4. |
| 16 | 02/11/23 | 09:00 - 11:00 | Correction of Homework 2. Countable union of countable sets. $\mathbb{Q}$ is countable. $\mathbb{R}$ and irrationals are uncountable. Complex numbers: Addition, multiplication. |
| 17 | 02/11/23 | 16:00 - 18:00 | Inverses. $\mathbb{C}$ is a field. $\mathbb{C}$ not ordered. Complex conjugate. Cartesian representation. Modulus. Triangle inequality. Trigonometric and exponential forms. |
| 18 | 03/11/23 | 09:00 - 10:00 | Correction of Tutorial 5. |
| 19 | 09/11/23 | 09:00 - 11:00 | Homework 2: went over most common mistakes. Canvas Announcements. Revision of Complex Numbers. Fundamental Theorem of algebra. Solving polynomial equations. |
| 20 | 09/11/23 | 16:00 - 18:00 | Polynomial division algorithm and examples. Roots of Unity. Roots in $\mathbb{C}$. |
| 21 | 10/11/23 | 09:00 - 10:00 | Correction of Tutorial 6. |
| 22 | 16/11/23 | 09:00 - 11:00 | Sequences: Definition and examples. Convergent sequences. Examples. Divergent sequences. Examples. Uniqueness of limit. |
| 23 | 16/11/23 | 16:00 - 18:00 | Bounded sequences. Convergent sequences are bounded. Converse is false: counterexample. Homework 3 Correction. |
| 24 | 17/11/23 | 09:00 - 10:00 | Correction of Tutorial 7. |
| 25 | 23/11/23 | 09:00 - 11:00 | Algebra of Limits Theorem. Examples. Fractional powers. Limit of square root of sequence. Examples. Squeeze Theorem and examples. |
| 26 | 23/11/23 | 16:00 - 18:00 | Correction of Homework 4. Geometric Sequence Test with proof. Examples. Ratio Test with proof. Examples. Examples in which Ratio test is inconclusive. |
| 27 | 24/11/23 | 09:00 - 10:00 | Correction of Tutorial 8. |
| 28 | 30/11/23 | 09:00 - 11:00 | Monotone sequences. Monotone Convergence Theorem. Euler’s Number. Sequences in $\mathbb{C}$. Boundedness. Algebra of Limits in $\mathbb{C}$. Examples. |
| 29 | 30/11/23 | 16:00 - 18:00 | Convergence to zero. GST and Ratio Test in $\mathbb{C}$. Convergence of Real and Imaginary parts. Series. Convergent Series. Necessary Condition. Telescopic series. |
| 30 | 01/12/23 | 09:00 - 10:00 | Examples. Geometric series and examples. |
| 31 | 07/12/23 | 09:00 - 11:00 | Algebra of Limits for series. Non-negative series: Cauchy Condensation Test, $p$-series, Comparison Test (CT), Limit CT, Ratio Test. General series. Absolute Convergence Test. |
| 32 | 07/12/23 | 16:00 - 18:00 | Ratio Test for general series. Exponential function and Euler’s Number. Conditional convergence. Riemann rearrangement Theorem. Dirichlet Test. Alternating Series Test. |
| 33 | 08/12/23 | 09:00 - 10:00 | Correction of Tutorial 9. |
| 34 | 14/12/23 | 09:00 - 11:00 | Correction of Tutorial 10. Solution of past Written Exams. |
| 35 | 14/12/23 | 16:00 - 18:00 | Revision and Exam Preparation. |
| 36 | 15/12/23 | 09:00 - 10:00 | Revision and Exam Preparation. |
Each week we have 1h of Tutorial in which I will solve exercises on the topics listed below. You should attempt solving the exercises before the tutorial.
| Date | Tutorial Number | Topics |
|---|---|---|
| 06/10/23 | 1 | Irrational numbers. |
| 13/10/23 | 2 | Basic set theory. Equivalence relation. |
| 20/10/23 | 3 | Absolute value. Triangle inequality. Induction. |
| 27/10/23 | 4 | Induction. Operations. Fields. |
| 03/11/23 | 5 | Fields. Sup and Inf. |
| 10/11/23 | 6 | Sup and inf. Inductive sets. Injectivity and surjectivity. Cardinality. |
| 17/11/23 | 7 | Complex numbers. Trigonometric and exponential form. Equations in $\mathbb{C}$. |
| 24/11/23 | 8 | Convergent sequences. Divergent sequences. Algebra of Limits. Limit Theorems. |
| 08/12/23 | 9 | Complex sequences. |
| 15/12/23 | 10 | Tests for convergence/divergence of series of real and complex numbers. |
Homework papers must be submitted on Canvas by 14:00 on the Due Date
Homework papers submitted outside of Canvas or after the deadline will NOT BE MARKED
Please submit PDFs only. Either:
| Due date | Homework Number | Topics |
|---|---|---|
| 11/10/23 | 1 | Irrational numbers. Basic set theory. |
| 25/10/23 | 2 | Order relation, Induction, Proofs. |
| 08/11/23 | 3 | Fields, Supremum and infimum. |
| 22/11/23 | 4 | Complex Numbers, Convergent sequences. |
| 15/12/23 | 5 | Convergence/Divergence of sequences and series. |