Theory and applications of infinite series

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Preface to the first edition......Page 6
Preface to the Second Edition......Page 8
Contents.\0......Page 10
Introduction\0......Page 14
1. The system of rational numbers and its gaps\0......Page 16
2. Sequences of rational numbers\0......Page 27
3 Irrational numbers\0......Page 36
4. Completeness and uniqueness of the system of real numbers\0......Page 46
5. Radix fractions and the Dedckznd section\0......Page 50
Exercises on Chapter 1 (1-8)\0......Page 55
6. Arbitrary sequences and arbitrary null sequences\0......Page 56
7. Powers, roots, and logarithms \'special null sequences\0......Page 62
8. Convergent sequences\0......Page 77
9 The two main criteria\0......Page 91
10 Limiting points and upper and lower limits\0......Page 102
11. Infinite series, infinite products, and infinite continued fractions\0......Page 111
Exercises on Chapter 11 (9-33)\0......Page 119
12. The first principal criterion and the two comparison tests\0......Page 123
13. The root test and the ratio test\0......Page 129
14 Series of positive, monotone decreasing terms\0......Page 133
Exercises on Chapter III (34-44)\0......Page 138
15. The second principal criterion and the algebra of convergent series\0......Page 139
16. Absolute convergence. Derangement of series\0......Page 149
17. Multiplication of infinite series\0......Page 159
Exercises on Chapter IV (45-63)\0......Page 162
18. The radius of convergence\0......Page 164
19. Functions of a real variable\0......Page 171
20. Principal pioperties of functions iepresented by power series\0......Page 184
21. The algebra of power series\0......Page 192
Exercises on Chapter V (64 -73)\0......Page 201
22. The rational functions\0......Page 202
23. The exponential function\0......Page 204
24. The trigonometrical functions\0......Page 211
25. The binomial series\0......Page 221
26. The logarithmic series\0......Page 224
27. The cyclometrical functions\0......Page 226
Exercises on Chapter VI (74 -84)\0......Page 228
28. Products with positive terms\0......Page 231
29. Products with arbitrary terms. Absolute conveigen e\0......Page 234
30. Connection between series and products. Conditional and unconditional convergence\0......Page 239
Exercises on Chapter VII (85-99)\0......Page 241
31. Statement of the problem\0......Page 243
32. Evaluation of the sum of a series by means of a closed expression\0......Page 245
33. Transformation of series\0......Page 253
34. Numerical evaluations\0......Page 260
35. Applications of the tranbiormation ofserics to numerical evaluations\0......Page 273
Exercises on Chapter VII[ (100-132)\0......Page 280
36. Detailed study of the two comparison tests\0......Page 287
37. The logarithmic scales\0......Page 291
38. Special comparison tests of the second kind\0......Page 297
39. Theorems of Abel, Dini, and Prin7sheim, and their application to a fresh deduction of the logarithmic scale of comparison tests\0......Page 303
40. Series of monotonely diminishing positive terms\0......Page 307
41. General remarks on the theory of the convergence and divergence of series of positive terms\0......Page 311
42. Systematization of the general theory of convergence\0......Page 318
Exercises on Chapter IX (133-141)\0......Page 324
43. Tests of convergence for series of arbitrary terms\0......Page 325
44. Rearrangement of conditionally convergent series\0......Page 331
45. Multiplication of conditionally convergent series\0......Page 333
Exercises on Chapter X (142--153)\0......Page 337
46. Uniform convergence\0......Page 339
47. Passage to the limit term by term\0......Page 351
48 Tests of uniform convergence\0......Page 357
A. Euler\'s formulae\0......Page 363
B. Dirachlet\'s integral\0......Page 369
C. Conditions of convergence\0......Page 377
50. Applications of the theory of Fourier series\0......Page 385
51. Products with variable terms\0......Page 393
Exercises on Chapter XI (154-173)\0......Page 398
52. Complex numbers and sequences\0......Page 401
53. Series of complex terms\0......Page 409
54. Power series. Analytic functions\0......Page 414
I. Rational functions\0......Page 423
II. The exponential function\0......Page 424
III. The functions cos z and sin z\0......Page 427
IV. The functions cotz and tanz\0......Page 430
V. The logarithmic series\0......Page 432
VI. The inverse sine series\0......Page 434
VII. The inverse tangent series\0......Page 435
VIII. The binomial series\0......Page 436
56. Series of variable terms. Uniform convergence. Weierstratis\' theorem on double series\0......Page 441
57. Products with complex terms\0......Page 447
A. Dirichlet\'s series\0......Page 454
B. Faculty series\0......Page 459
C. Lambert\'s series\0......Page 461
Exercises on Chapter XII (174-199)\0......Page 465
59. General remarks on divergent series and the processes of limitation\0......Page 470
60. The C- and H- processes\0......Page 491
61. Application of C1- summation to the theory of Fourier series\0......Page 505
62. The A- process\0......Page 511
63. The E- process\0......Page 520
Exercises on Chapter XIII (200-216)\0......Page 529
64. Euler\'s summation formula\0......Page 0
A. The summation formula\0......Page 531
B. Applications\0......Page 538
C. The evaluation of  remiindcrs\0......Page 544
65. Asymptotic series\0......Page 548
A. Examples of the expansion problem\0......Page 556
B. Examples of the summation problem\0......Page 561
Exercises on Chapter XIV (217-225)\0......Page 566
Bibliography\0......Page 569
Name and subject index\0......Page 570