Sequences and Series: contents

1. Sequences and Series Home page

The "home" for these pages on Sequences and Series is at http://web.mat.bham.ac.uk/R.W.Kaye/seqser/. You will find there a preface, as well as legal details (such as the rules you must follow if you want to copy any part of this text).

  1. Home page and preface.
  2. Contents, this page.
  3. The MathML used in these pages and notes for setting up your web browser if required.

2. Background

  1. The Greek Alphabet.
  2. A glossary of terms.
  3. Number systems.
  4. Mathematical Induction.

3. Sequences

  1. Sequences and series: an introduction.
  2. The triangle inequality.
  3. Why do we need to do analysis rigorously?
  4. The definition of convergence.
  5. Logic for analysis: statements.
  6. Logic for analysis: proofs.
  7. Examples of convergent sequences.
  8. Sequences that are eventually constant.
  9. Theorem on uniqueness of limits.
  10. Theorems on bounds.
  11. Subsequences.
  12. Using subsequences to prove convergence.
  13. Sums of sequences.
  14. Products of sequences.
  15. Quotients of sequences.
  16. Continuity.
  17. Archimedean Ordered Fields.
  18. Density results for the rationals.
  19. Properties of monotone sequences.
  20. The Monotone Convergence Theorem and Completeness of the Reals.
  21. Applications of monotone convergence to roots.
  22. The arithmetic-mean/geometric-mean inequality.
  23. The Euler number e.
  24. The natural logarithm.
  25. The Bolzano-Weierstrass Theorem.
  26. Cauchy Sequences.
  27. Supremum and infimum.
  28. Fekete's lemma.

4. Series

  1. Introduction to infinite series.
  2. Some standard convergent series.
  3. The comparison tests.
  4. Applications of the comparison tests.
  5. The ratio test.
  6. Alternating series.
  7. Absolute convergence.
  8. Power Series.

5. Number systems

  1. Introduction to constructions of number systems.
  2. Construction of the natural numbers.
  3. Construction of the integers.
  4. Construction of the rationals.
  5. Construction of the reals via monotonic sequences.
  6. Construction of the reals via Cauchy sequences.
  7. Construction of the reals via Dedekind cuts.
  8. Construction of the reals via almost linear maps.
  9. The uniqueness of the reals.

6. Exercises

  1. Exercise sheet 1.
  2. Exercise sheet 2.
  3. Exercise sheet 3.
  4. Exercise sheet 4.
  5. Exercise sheet 5.
  6. Exercise sheet 6.
  7. Exercise sheet 7.
  8. Exercise sheet 8.
  9. Exercise sheet 9.
  10. Exercise sheet 10.

 Richard Kaye. 2015-09-30. Copyright Richard Kaye, http://web.mat.bham.ac.uk/R.W.Kaye/.