Analysis I: Sequences and Series -- Dr Stewart -- 14 MT
In these lectures we introduce the real and complex numbers and study
their properties, particularly completeness; define and study limits of
sequences, convergence of series, and power series.
Real numbers: arithmetic, ordering, suprema, infima;
the real numbers as a complete ordered field.
The reals are uncountable.
The complex number system. The triangle inequality.
Sequences of real or complex numbers.
Definition of a limit of a sequence of numbers.
Limits and inequalities. The algebra of limits. Order notation: O, o.
Subsequences; a proof that
every subsequence of a convergent sequence converges to the same limit;
bounded monotone sequences converge. Bolzano-Weierstrass Theorem. Limit
point of a set. Cauchy's convergence principle.
Series of real or complex numbers. Convergence of series.
Simple examples to include geometric progressions and
some power series. Absolute convergence, comparison test, ratio test,
integral test. Alternating series test.
Power series, radius of convergence;
important examples to include the exponential, cosine and sine series.
Main texts
Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis,
Third Edition(2000), Wiley. Chapters 2, 3, 9.1, 9.2.
[Please note: This book is available at approximately half its list
price from Blackwells in Oxford.]
R. P. Burn, Numbers and Functions, Steps into Analysis,
Cambridge University Press (2000), Chapters 2-6.
(This is a book of problems and answers, a DIY course in analysis.)
Alternative reading
The first five books take a slightly gentler approach to the material
in the syllabus, whereas, the last two cover it in greater depth and
contain some more advanced material.
Mary Hart, A Guide to Analysis, MacMillan (1990), Chapter 2.
J. C. Burkill, A First Course In Mathematical Analysis, CUP (1962),
Chapters 1, 2 and 5.
K. G. Binmore, Mathematical Analysis, A Straightforward Approach,
Cambridge University Press, Chapters, 1-6.
Victor Bryant, Yet Another Introduction to Analysis,
Cambridge University Press (1990), Chapters 1 and 2.
G. Smith, Introductory Mathematics: Algebra and Analysis,
Springer-Verlag (1998), Chapter 3 (introducing complex numbers).
Michael Spivak, Calculus, Benjamin (1967), Parts I, IV, and V
(for a construction of the real numbers).
Brian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner,
Elementary Analysis, Prentice Hall (2001), Chapters 1-4.