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

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## James M. Hyslop:

## Contents

## James M. Hyslop: *Infinite Series (3rd Edition)*

Published $1947$, **Oliver and Boyd**.

### Subject Matter

### Contents

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

- Preface (R.A.F. Middle East

- 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

## Further Editions

## Source work progress

- 1947: James M. Hyslop:
*Infinite Series*(3rd ed.) ... (previous): Chapter $\text I$: Functions and Limits: $\S 3$: Bounds of a Function: Theorem $\text{A}$

- There are some examples not processed
- Still some work to be done structuring 1942: James M. Hyslop:
*Infinite Series*which has some entries not yet included in the source flow