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

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

Published $1947$, Oliver and Boyd.

### 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

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