# MA137 Content

**Aims:
**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.

**:**

Content

Content

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

**Objectives:**

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

**Books**:

(Recommended)

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.