MA137-24 Mathematical Analysis
Many problems in mathematics cannot be solved explicitly. So one resorts to finding approximate solutions and estimate the error between a true solution and the approximate one. Indeed, one may even be able to demonstrate the existence of a solution by exhibiting a sequence of approximate solutions that converge to an exact solution.
The study of limiting processes is the central theme in mathematical analysis. It involves the quantification of the notion of limit and precise formulation of intuitive notions of infinite sums, functions, continuity and the calculus.
You will study ideas of the mathematicians Cauchy, Dirichlet, Weierstrass, Bolzano, D'Alembert, Riemann and others, concerning sequences and series in term one, continuity and differentiability in term two.
By the end of the module the student should be able to:
Understand what is meant by the symbol 'infinity'
Understand what it means for a sequence to converge or diverge and to compute simple limits
Determine when it makes sense to add up infinitely many numbers
Understand the notions of continuity and differentiability
Establish various properties of continuous and differentiable functions
Answer the question "when can a function be represented by a power series?"
Develop their own methods for solving problems
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
- Decimal expressions and real numbers; the geometric series and conversion of recurring decimals into fractions.
- Convergence of a nonrecurring decimal and the completeness axiom in the form that an increasing sequence which is bounded above converges to a real number.
- The completeness axiom as the main distinguishing feature between the rationals and the reals; approxiamtion of irrationals by rationals and vice-versa.
- Formal definition of sequence and subsequence.
- Limit of a sequence of real numbers; Cauchy sequences and the Cauchy criterion.
(a) Series with positive terms
(b) Alternating series
- The number e both as lim(1+(1/n))^n and as 1 + 1 + (1/2!) + (1/3!) + ... .
- Bounded and unbounded sets. Sups and infs.
- Properties of continuous functions
- Continuous Limits
- Properties of differentiable functions
- Higher order derivatives
- Power Series
- Taylor’s Theorem
- The Classical Functions of Analysis
- Upper and Lower Limits
By the end of the module, students should be able to:
- Understand what is meant by the symbol `infinity'
- Understand what it means for a sequence to converge or diverge and to compute simple limits
- Determine when it makes sense to add up infinitely many numbers
- Understand the notions of continuity and differentiability
- Establish various properties of continuous and differentiable functions
- Answer the question "when can a function be represented by a power series?"
- Develop their own methods for solving problems
Indicative reading list
D. Stirling, Mathematical Analysis and Proof, 1997.
M. Spivak, Calculus, Benjamin.
M. Hart, Guide to Analysis, Macmillan. (A good traditional text with theory and many exercises.)
G.H. Hardy, An introduction to Pure Mathematics, CUP.
Subject specific skills
See learning outcomes
Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.
|Lectures||60 sessions of 1 hour (25%)|
|Seminars||18 sessions of 30 minutes (4%)|
|Private study||171 hours (71%)|
Private study description
171 hours private study, revision for exams, and non-assessed assignments
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Students can register for this module without taking any assessment.
Assessment group D3
|3 hour online examination (Summer)||60%|
|Term 1 Exam||25%|
Assessment group R
Feedback on assessment
Assignments marked by supervisors, typically returned within one week.
This module is Core for:
- Year 1 of USTA-G302 Undergraduate Data Science
- Year 1 of USTA-G304 Undergraduate Data Science (MSci)
- Year 1 of UCSA-G4G1 Undergraduate Discrete Mathematics
- Year 1 of UCSA-G4G3 Undergraduate Discrete Mathematics
- Year 1 of USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
- Year 1 of UMAA-GV17 Undergraduate Mathematics and Philosophy
- Year 1 of UPXA-FG33 Undergraduate Mathematics and Physics (BSc MMathPhys)
- Year 1 of UPXA-GF13 Undergraduate Mathematics and Physics (BSc)
- Year 1 of UPXA-FG31 Undergraduate Mathematics and Physics (MMathPhys)
- Year 1 of USTA-G1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
- Year 1 of USTA-GG14 Undergraduate Mathematics and Statistics (BSc)
- Year 1 of USTA-Y602 Undergraduate Mathematics,Operational Research,Statistics and Economics