Webpage of the module 400297 T1 2024/25
Welcome to the module Numbers, Sequences and Series 400297 for the BSc in Mathematics at the University of Hull, academic year 2024/25. This is an introductory module in Mathematical Analysis. We will investigate the properties of real numbers and explore the concepts of limit and series.
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All the module information will be posted on this page, as well as on Canvas. The links to the reference material are:
Each week we have:
Times and venues are as follows:
Lecture 1: Thursday 9:00-11:00 in Robert Blackburn Building - Lecture Theatre D
Please check on MyTimetable every week to see if there are any changes to the following session times and rooms.
This module will be assessed as follows:
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 module Lecture Notes are available here. Topics covered in each lecture are as follows:
| Class # | Date | Time | Topics |
|---|---|---|---|
| 1 | 26/09/24 | 09:00 - 11:00 | Canvas, Lecture Notes, Assessment etc. Introduction: Numbers $\mathbb{N}, \mathbb{Z}, \mathbb{Q}$ and $\mathbb{R}$. Proof: $\sqrt{2} \notin \mathbb{Q}$. |
| 2 | 26/09/24 | 12:00 - 14:00 | Preliminaries: Sets, Basic Logic, Operations on Sets, infinite unions and intersections. Examples. Exercises in Tutorial 1. |
| 3 | 27/09/24 | 13:00 - 14:00 | Finished exercises in Tutorial 1. Complement, Power set. Examples. |
| 4 | 03/10/24 | 09:00 - 11:00 | Product of sets. Binary relations. Equivalence, Partial Order, Total order relations. Intervals. Functions. |
| 5 | 03/10/24 | 12:00 - 14:00 | Exercises in Tutorial 2. Absolute value. Geometric meaning. Basic lemma on absolute value. Triangle inequality (with proof). Corollaries of Triangle inequality. |
| 6 | 04/10/24 | 13:00 - 14:00 | Proofs in Mathematics. Induction. Recurrence sequence. Examples. Hints for Homework 1. |
| 7 | 10/10/24 | 09:00 - 11:00 | Binary operations. Commutativity, associativity, neutral element, inverse. Fields. Examples. Ordered Fields. $\mathbb{Q}$ is ordered field. |
| 8 | 10/10/24 | 12:00 - 14:00 | Partition of a set, Cut of a set, Cut Property. Proof that $\mathbb{Q}$ does not have the Cut Property. Upper bound, bounded above, sup. Uniqueness of sup. Maximum. |
| 9 | 11/10/24 | 13:00 - 14:00 | Exercises in Tutorial 3. Correction of Homework 1. |
| 10 | 17/10/24 | 09:00 - 11:00 | Lower bound, inf, min. $\mathbb{Q}$ is not complete (partial proof). Axiom of Completeness. Definition of $\mathbb{R}$. Theorem: $\mathbb{R}$ exists unique (no proof). |
| 11 | 17/10/24 | 12:00 - 14:00 | $\mathbb{N}$ as inductive subset of $\mathbb{R}$. Properties. Principle of Induction. Exercises in Tutorial 4. |
| 12 | 18/10/24 | 13:00 - 14:00 | Definition of $\mathbb{Z}$, $\mathbb{Q}$ and properties. Archimedean property 1 (proof) and 2 (no proof). Nested Interval Property (proof). Counterexample with open intervals. |
| 13 | 24/10/24 | 09:00 - 11:00 | Examples: Calculation of sup and inf. Existence of $k$-th roots (no proof). $\mathbb{Q}$ and irrationals dense in $\mathbb{R}$. Injectivity, Surjectivity, Bijectivity. |
| 14 | 24/10/24 | 12:00 - 14:00 | Exercises in Tutorial 5. Correction of Homework 2. Cardinality. Subsets of countable sets. Examples. $\mathbb{N}$, $\mathbb{Z}$ are countable. |
| 15 | 25/10/24 | 13:00 - 14:00 | Countable unions. $\mathbb{Q}$ countable. $\mathbb{R}$ and Irrationals uncountable. Complex numbers: Addition, multiplication, Inverses. $\mathbb{C}$ is a field (not ordered). |
| 16 | 31/10/24 | 09:00 - 11:00 | Complex conjugate. Cartesian representation. Modulus. Triangle inequality. Trigonometric and exponential forms. |
| 17 | 31/10/24 | 12:00 - 14:00 | Fundamental Theorem of algebra. Polynomial division algorithm and examples. Exercises in Tutorial 6. |
| 18 | 01/11/24 | 13:00 - 14:00 | Solving polynomial equations in $\mathbb{C}$: order 2, and higher order. |
| 19 | 07/11/24 | 09:00 - 11:00 | |
| 20 | 07/11/24 | 12:00 - 14:00 | Roots of Unity. Roots in $\mathbb{C}$. Sequences: Definition and examples. Convergent sequences. Examples. Divergent sequences. Examples. Uniqueness of limit. |
| 21 | 08/11/24 | 13:00 - 14:00 | Exercises in Tutorial 7. |
| 22 | 14/11/24 | 09:00 - 11:00 | |
| 23 | 14/11/24 | 12:00 - 14:00 | Bounded sequences. Convergent sequences are bounded. Converse is false: counterexample. Homework 3 Correction. |
| 24 | 15/11/24 | 13:00 - 14:00 | Exercises in Tutorial 8. |
| 25 | 21/11/24 | 09:00 - 11:00 | Algebra of Limits Theorem. Examples. Fractional powers. Limit of square root of sequence. Examples. Squeeze Theorem and examples. |
| 26 | 21/11/24 | 12:00 - 14:00 | Correction of Homework 4. Geometric Sequence Test with proof. Examples. Ratio Test with proof. Examples. Examples in which Ratio test is inconclusive. |
| 27 | 22/11/24 | 13:00 - 14:00 | Exercises in Tutorial 9. |
| 28 | 28/11/24 | 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 | 28/11/24 | 12:00 - 14:00 | Convergence to zero. GST and Ratio Test in $\mathbb{C}$. Convergence of Real and Imaginary parts. Series. Convergent Series. Necessary Condition. Telescopic series. |
| 30 | 29/12/24 | 13:00 - 14:00 | Examples. Geometric series and examples. |
| 31 | 05/12/24 | 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 | 05/12/24 | 12:00 - 14:00 | Ratio Test for general series. Exponential function and Euler’s Number. Conditional convergence. Riemann rearrangement Theorem. Dirichlet Test. Alternating Series Test. |
| 33 | 06/12/24 | 13:00 - 14:00 | Exercises in Tutorial 10. |
| 34 | 12/12/24 | 09:00 - 11:00 | Solution of Exam papers 2018/2019, 2019/20, 2023/24. |
| 35 | 12/12/24 | 12:00 - 14:00 | Revision and Exam Preparation. |
| 36 | 13/12/24 | 13:00 - 14:00 | Revision and Exam Preparation. |
Each week we have 1h of Tutorial in which we will solve exercises on the topics listed below. You should attempt solving the exercises before the tutorial.
| Date | Tutorial # | Topics |
|---|---|---|
| 26/09/24 | 1 | Irrational numbers. |
| 03/10/24 | 2 | Irrational numbers. Basic set theory. |
| 10/10/24 | 3 | Equivalence relation. Absolute value. Triangle inequality. Induction. |
| 17/10/24 | 4 | Induction. Operations. Fields. |
| 24/10/24 | 5 | Sup and inf. Inductive sets. |
| 31/10/24 | 6 | Injectivity and surjectivity. Cardinality. Complex numbers. Equations in $\mathbb{C}$. |
| 07/11/24 | 7 | Convergent sequences. Divergent sequences. Algebra of Limits. Limit Theorems. |
| 14/11/24 | 8 | Complex sequences. |
| 21/11/24 | 9 | Geometric Series, Cauchy Condensation, Comparison, Limit Comparison, Ratio Tests. |
| 28/12/24 | 10 | Absolute and conditional convergence. |
There will be 5 Homework papers in total:
Each homework paper:
| Due date | Homework # | Topics |
|---|---|---|
| 08/10/24 | 1 | Irrational numbers. Basic set theory. |
| 22/10/24 | 2 | Order relation, Induction, Proofs. |
| 05/11/24 | 3 | Fields, Supremum and infimum. |
| 19/11/24 | 4 | Complex Numbers, Convergent sequences. |
| 03/12/24 | 5 | Convergence/Divergence of sequences and series. |
Homework papers submitted outside of Canvas or more than 24 hours after the Due Date will NOT BE MARKED
Please submit PDFs only. Either: