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 web page

Module aims

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.

1. Decimal expressions and real numbers; the geometric series and conversion of recurring decimals into fractions.
2. 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.
3. The completeness axiom as the main distinguishing feature between the rationals and the reals; approxiamtion of irrationals by rationals and vice-versa.
4. Inequalities.
5. Formal definition of sequence and subsequence.
6. Limit of a sequence of real numbers; Cauchy sequences and the Cauchy criterion.
7. Series:
   (a) Series with positive terms
   (b) Alternating series
8. The number e both as lim(1+(1/n))^n and as 1 + 1 + (1/2!) + (1/3!) + ... .
9. Bounded and unbounded sets. Sups and infs.
10. Continuity
11. Properties of continuous functions
12. Continuous Limits
13. Differentiability
14. Properties of differentiable functions
15. Higher order derivatives
16. Power Series
17. Taylor’s Theorem
18. The Classical Functions of Analysis
19. 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.