## Syllabus

From the syllabus of the course we could have read that the course was to introduce students to a mathematically rigorous treatment of the most basic concepts of analysis. Thereby laying the foundations for further study of the subject and developing the understanding of mathematical rigour. As applaudable a goal now as it was then.

The goal of the course was to cover the following. Properties of the Real Number System, Sequences of Real Numbers, Series of Real Numbers, Comparison and Ratio Tests for Series of Positive Terms, Series with Positive and Negative Terms, Alternating Series Test. Convergence of Power Series, Radius of Convergence, Cauchy's Root Test, Functions Defined by Power Series.

## Lecture Notes

Notes Contents
Chapter 1 Properties of the real number system. Inequalities.
Chapter 2 Bounded and unbounded sets. Least upper bound and greatest lower bound.
Chapter 3 Sequences of real numbers. Subsequences. Limit of a sequence. Convergence and divergence. Convergent sequences are bounded. A monotonic increasing sequence which is bounded above is convergent.
Chapter 4 Sums, products and quotients of convergent sequences. The sequences {nk} and {rn}. Sandwich rule.
Chapter 5 Series of real numbers as sequences of partial sums. Sums of Geometric series. Divergence of series.
Chapter 6 Comparison and ratio tests for series of positive terms.
Chapter 7 Series with positive and negative terms. Alternating series test. Relative and absolute convergence. Convergence of power series. Radius of convergence. Cauchy's root test. Functions defined by power series (trigonometric and exponential functions).

## Question Sheets

Additional examples for you to practice on are available here:

These questions build on, and develop further some of the questions in the six example sheets above.

Unfortunately there are no typed answers for any of the questions in this course.