Lancaster University

Sequences and series: formal proofs of the basic results. Every bounded sequence has a convergent subsequence.

Limits and continuity of functions. The definitions. Sums, products, compositions, inverse functions.

Differentiation: definition; sums, products, compositions, inverse functions. L'Hopital's rule. Power series.

Consequences of continuity on an interval: intermediate value theorem, boundedness, uniform continuity.

- Limits of sequences: basic results; monotonic sequences; subsequences.
- Infinite series: standard examples; comparison and ratio tests; absolute convergence; power series; Abel summation; double series.
- Limits and continuity of functions.
- Differentiation: the definition and basic results; compositions and inverse functions; differentiation of power series.
- Intermediate value theorem. Boundedness and uniform continuity of functions continuous on a closed interval.
- The mean value theorem; applications to identities and inequalities.
- Definition of the Riemann integral. The fundamental theorem of calculus.
- Inequalities for integrals; application to the estimation of discrete sums; the series for tan -1 x and log (1+ x); Euler's constant.
- Infinite products.
- Sequences and series of functions: uniform convergence.
- Fourier series: examples, convergence theorems and applications.

Prerequisites:  MATH101 Calculus; MATH115 Series and Functions

The notion of a limit underlies a whole range of concepts that are really basic in mathematics, including sums of infinite series, continuity, differentiation and integration. After the more informal treatment in the first year, our aim now is to develop a really precise understanding of these notions and to provide fully watertight proofs of the theorems involving them. We also show how the theorems apply to give useful facts about specific functions such as exp, log, sin, cos, including some integrals and other unexpected identities.

R G Bartle and D R Sherbert, Introduction to Real Analysis, Wiley, 1982.
K G Binmore, Mathematical Analysis (second edition), Cambridge University Press, 1982.
J B Reade, An Introduction to Mathematical Analysis, Clarendon, 1986.
M Spivak, Calculus, Benjamin, 1967.
D S G Stirling, Mathematical Analysis, Ellis Horwood, 1987.
G H Hardy, Pure Mathematics (tenth edition), Cambridge University Press, 1963.