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.
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; approximation of irrationals by rationals and vice-versa.
5. Formal definition of sequence and subsequence.
6. Limit of a sequence of real numbers; Cauchy sequences and the Cauchy criterion.
(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.
11. Properties of continuous functions
12. Continuous Limits
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
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
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, A Course of Pure Mathematics, CUP.