Book:James M. Hyslop/Infinite Series/Third Edition

Subject Matter

 * Analysis

Contents

 * Preface (R.A.F. Middle East August 1942)
 * Preface to Second Edition (Girvan January 1945)


 * Chapter $\text {I}$: Functions and Limits


 * 1. Introduction
 * 2. Functions
 * 3. Bounds of a Function
 * 4. Limits of Functions
 * 5. Two Important Limits
 * 6. Monotonic Functions
 * 7. Upper and Lower Limits
 * 8. Continuity
 * 9. Differentiation
 * 10. Integration
 * 11. The $o$, $O$ notation
 * Examples


 * Chapter $\text {II}$: Some Properties of Particular Functions


 * 12. The Logarithmic and Exponential Functions
 * 13. The Hyperbolic Functions
 * 14. The Circular Functions
 * Examples


 * Chapter $\text {III}$: Real Sequences and Series


 * 15. Definition of a Sequence
 * 16. Convergent, Divergent and Oscillating Sequences
 * 17. Infinite Series
 * 18. Important Particular Series
 * 19. The General Principle of Convergence
 * 20. Some Preliminary Theorems on Series
 * Examples


 * Chapter $\text {IV}$: Series of Non-Negative Terms


 * 21. A Fundamental Theorem
 * 22. Rearrangement of Terms
 * 23. Tests for Convergence
 * 24. The Integral Test
 * 25. The Comparison Test
 * 26. The Ratio or d'Alembert's Test
 * 27. Cauchy's Condensation Test
 * 28. Connection between the Ratio Test and Cauchy's Test
 * 29. A General Test for Convergence
 * 30. Raabe's Test
 * 31. Gauss's Test
 * 32. Euler Constant
 * 33. Stirling's Approximation for $n!$


 * Chapter $\text {V}$: General Series


 * 34. Real Series
 * 35. Absolute Convergence
 * 36. Tests for Absolute Convergence
 * 37. Conditional Convergence
 * 38. Riemann's Theorem
 * 39. Complex Limits
 * 40. Series whose Terms may be Complex
 * 41. Abel's Lemma
 * Examples


 * Chapter $\text {VI}$: Series of Functions


 * 42. Uniform Convergence
 * 43. Series of Functions
 * 44. Tests for Uniform Convergence
 * 45. Some Properties of Uniformly Convergent Series
 * 46. Power Series
 * Examples


 * Chapter $\text {VII}$: The Multiplication of Series


 * 47. Multiplication of Series of Non-Negative Terms
 * 48. Multiplication of General Series
 * Examples


 * Chapter $\text {VIII}$: Infinite Products


 * 49. Convergence and Divergence of Infinite Products
 * 50. Some Theorems on Special Types of Products
 * 51. The Absolute Convergence of Infinite Products
 * 52. The Uniform Convergence of an Infinite Product
 * 53. The Infinite Products for $\sin x$ and $\cos x$
 * 54. The Gamma Distribution
 * Examples


 * Chapter $\text {IX}$: Double Series


 * 55. Introduction
 * 56. Double Series whose Terms are Non-Negative
 * 57. The Absolute Convergence of a Double Series
 * 58. The Interchange of the Order of Summation for Repeated Series
 * Examples


 * Index



Source work progress
* : Chapter $\text I$: Functions and Limits: $\S 4$: Limits of Functions: Theorem $1 \ \text{(iii)}$


 * Note that there is considerable refactoring needed around the Combination Theorem for Limits of Functions.