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Time and place: Tue 17-19 in L3; Thu 16-17 in MS.02.

Assessment: Material from this half of the course is examined in the first week of Term 2, probably Monday.

Description:

Inequalities: rules for manipulating inequalities; inequalities and powers; inequalities and absolute values; Bernoulli's inequality; triangle inequality.

Sequences: monotonic sequences; bounded sequences; subsequences; tending to infinity; null sequences (and algebra of); convergent sequences (and algebra of, boundedness, uniqueness of limit); sandwich theorems; shift rules; standard limits; ratio test; limits and inequalities; recursively-defined sequences.

The real numbers: infinitely many rationals/irrationals in any open interval; rationals/irrationals and terminating/recurring/non-recurring decimals; numbers with more than one decimal representation; sets and upper/lower bounds; supremum and infimum; completeness axiom (in the form "every non-empty set bounded above has a supremum"); consequences of completeness; existence of k-th roots; Bolzano-Weierstrass theorem; Cauchy sequences (contracting sequences as example).

Series: partial sums; convergence and divergence; sum rule; shift rule; null sequence test.

- Series with positive terms: boundedness condition, comparison tests, ratio test, integral test.
- Alternating series: alternating series test, absolute convergence, absolute value form of ratio test, conditional convergence, rearrangements.

**Module material**:

I occasionally put comments on my facebook page; but anything of importance can be found here.

Week 1, assignment due 11 October

Week 2, assignment due 18 October

Week 3, assignment due 25 October

Week 4, assignment due 1 November

Week 5, assignment due 8 November

Week 6, assignment due 15 November

Week 7, assignment due 22 November

Week 8, assignment due 29 November

Week 9, assignment due 6 December

**References**:

Mary Hart,

*Guide to Analysis*, Macmillan Mathematical Guides, 2001, ISBN: 0333794494.G. H. Hardy,

*A Course of Pure Mathematics*, Cambridge Mathematical Library, 1993, ISBN: 0521092272.Michael Spivak,

*Calculus*, Publish Or Perish, 1994, ISBN: 0914098896.David S. G. Stirling,

*Mathematics Analysis and Proof*, Horwood Publishing, 1997, ISBN: 1898563365.