MA13724 Mathematical Analysis
Introductory description
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.
Module aims
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
Outline syllabus
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 viceversa.
 Inequalities.
 Formal definition of sequence and subsequence.
 Limit of a sequence of real numbers; Cauchy sequences and the Cauchy criterion.
 Series:
(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.
 Continuity
 Properties of continuous functions
 Continuous Limits
 Differentiability
 Properties of differentiable functions
 Higher order derivatives
 Power Series
 Taylor’s Theorem
 The Classical Functions of Analysis
 Upper and Lower Limits
Learning outcomes
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
Transferable skills
Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.
Study time
Type  Required 

Lectures  60 sessions of 1 hour (25%) 
Seminars  18 sessions of 30 minutes (4%) 
Private study  171 hours (71%) 
Total  240 hours 
Private study description
171 hours private study, revision for exams, and nonassessed assignments
Costs
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
Weighting  Study time  

Weekly Assignments  15%  
3 hour online examination (Summer)  60%  


Term 1 Exam  25% 
Assessment group R
Weighting  Study time  

Exam  100%  
exam

Feedback on assessment
Assignments marked by supervisors, typically returned within one week.
Courses
This module is Core for:
 Year 1 of USTAG302 Undergraduate Data Science
 Year 1 of USTAG304 Undergraduate Data Science (MSci)
 Year 1 of UCSAG4G1 Undergraduate Discrete Mathematics
 Year 1 of UCSAG4G3 Undergraduate Discrete Mathematics
 Year 1 of USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
 Year 1 of UMAAGV17 Undergraduate Mathematics and Philosophy
 Year 1 of UPXAFG33 Undergraduate Mathematics and Physics (BSc MMathPhys)
 Year 1 of UPXAGF13 Undergraduate Mathematics and Physics (BSc)
 Year 1 of UPXAFG31 Undergraduate Mathematics and Physics (MMathPhys)
 Year 1 of USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
 Year 1 of USTAGG14 Undergraduate Mathematics and Statistics (BSc)
 Year 1 of USTAY602 Undergraduate Mathematics,Operational Research,Statistics and Economics