Book:J.A. Green/Sequences and Series

Subject Matter

 * Real Analysis

Contents

 * Preface


 * Chapter $1$. Sequences
 * 1. Infinite sequences
 * 2. Successive approximations
 * 3. Graphical representation of a sequence
 * 4. The limit of a sequence
 * 5. Other types of sequence
 * 6. Rules for calculating limits
 * 7. Some dangerous expressions
 * 8. Subsequences
 * 9. Monotone sequences and bounded sequences
 * 10. The functions $x^n$, $n^s$ and $n^s x^n$
 * 11. Solution of equations by iteration
 * Exercises


 * Chapter $2$. Infinite series
 * 1. Finite series
 * 2. Infinite series
 * 3. Convergent and divergent series
 * 4. Some examples of infinite series
 * 5. Some rules for convergent series
 * 6. A test for divergence
 * 7. The comparison test
 * 8. The ratio test
 * 9. The integral test
 * 10.  Series with positive and negative terms. Leibniz's test
 * 11. Absolute convergence
 * 12. Power series
 * 13. Multiplication of series
 * 14. Notes on the use of the convergence tests
 * Exercises


 * Chapter 3. Further techniques and results
 * 1. Numerical calculation of the sum of a series
 * 2. Estimating the remainder of a power series
 * 3. Integration of power series
 * 4. Differentiation of power series
 * 5. Cauchy's convergence principle
 * 6. Dirichlet's convergence test
 * Exercises


 * Answers to Exercises


 * Index



Source work progress

 * : Chapter $1$: Sequences: $1$. Infinite Sequences: Example $4$