9780486661650
Theory And Application Of Infinite Series - Konrad Knopp
Dover Publications (1990)
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Series, Infinite

Unusually clear and interesting classic covers real numbers and sequences, foundations of the theory of infinite series and development of the theory (series of valuable terms, Euler's summation formula, asymptotic expansions, other topics). Exercises throughout. Ideal for self-study.

Product Details
LoC Classification QA295 .K74 1990
Dewey 515.243
Format Paperback
Cover Price 21,95 €
No. of Pages 563
Height x Width 215 x 137 mm

Notes
Contents.
Introduction
Part I. Real numbers and sequences 1
Chapter 1. Principles of the theory of real numbers 1
1. The system of rational numbers and its gaps 1
2. Sequences of rational numbers 14
3 Irrational numbers 23
4. Completeness and uniqueness of the system of real numbers 33
5. Radix fractions and the Dedckznd section 37
Exercises on Chapter 1 (1-8) 42
Chapter II. Sequences of real numbers. 43
6. Arbitrary sequences and arbitrary null sequences 43
7. Powers, roots, and logarithms 'special null sequences 49
8. Convergent sequences 64
9 The two main criteria 78
10 Limiting points and upper and lower limits 89
11. Infinite series, infinite products, and infinite continued fractions 98
Exercises on Chapter 11 (9-33) 106
Part II. Foundations of the theory of infinite series.
Chapter III. Series of positive terms 110
12. The first principal criterion and the two comparison tests 110
13. The root test and the ratio test 116
14 Series of positive, monotone decreasing terms 120
Exercises on Chapter III (34-44) 125
Chapter IV. Page Series of arbitrary terms. 126
15. The second principal criterion and the algebra of convergent series 126
16. Absolute convergence. Derangement of series 136
17. Multiplication of infinite series 146
Exercises on Chapter IV (45-63) 149
Chapter V. Power series. 151
18. The radius of convergence 151
19. Functions of a real variable 158
20. Principal pioperties of functions iepresented by power series 171
21. The algebra of power series 179
Exercises on Chapter V (64 -73) 188
Chapter VI The expansions of the so-called elementary functions. 189
22. The rational functions 189
23. The exponential function 191
24. The trigonometrical functions 198
25. The binomial series 208
26. The logarithmic series 211
27. The cyclometrical functions 213
Exercises on Chapter VI (74 -84) 215
Chapter VII. Infinite products. 218
28. Products with positive terms 218
29. Products with arbitrary terms. Absolute conveigen e 221
30. Connection between series and products. Conditional and unconditional convergence 226
Exercises on Chapter VII (85-99) 228
Chapter VI[I. 2 Closed and numerical expressions for the sums of series. 230
31. Statement of the problem 230
32. Evaluation of the sum of a series by means of a closed expression 232
33. Transformation of series 240
34. Numerical evaluations 247
35. Applications of the tranbiormation ofserics to numerical evaluations 260
Exercises on Chapter VII[ (100-132) 267
Part III Development of the theory. 274
Chapter IX. Series of positive terms. 274
36. Detailed study of the two comparison tests 274
37. The logarithmic scales 278
38. Special comparison tests of the second kind 284
39. Theorems of Abel, Dini, and Prin7sheim, and their application to a fresh deduction of the logarithmic scale of comparison tests 290
40. Series of monotonely diminishing positive terms 294
41. General remarks on the theory of the convergence and divergence of series of positive terms 298
42. Systematization of the general theory of convergence 305
Exercises on Chapter IX (133-141) 311
Chapter X. Series of arbitrary terms. 312
43. Tests of convergence for series of arbitrary terms 312
44. Rearrangement of conditionally convergent series 318
45. Multiplication of conditionally convergent series 320
Exercises on Chapter X (142--153) 324
Chapter XI. Series of variable terms (Sequences of functions). 326
46. Uniform convergence 326
47. Passage to the limit term by term 338
48 Tests of uniform convergence 344
49. Fourier series 350
A. Euler's formulae 350
B. Dirachlet's integral 356
C. Conditions of convergence 364
50. Applications of the theory of Fourier series 372
51. Products with variable terms 380
Exercises on Chapter XI (154-173) 385
Chapter XlI. Series of complex terms. 388
52. Complex numbers and sequences 388
53. Series of complex terms 396
54. Power series. Analytic functions 401
55. The elementary analytic functions 410
I. Rational functions 410
II. The exponential function 411
III. The functions cos z and sin z 414
IV. The functions cot z and tan z 417
V. The logarithmic series 419
VI. The inverse sine series 421
VII. The inverse tangent series 422
VIII. The binomial series 423
56. Series of variable terms. Uniform convergence. Weierstratis' theorem on double series 428
57. Products with complex terms 434
58. Special classes of series of analytic functions 441
A. Dirichlet's series 441
B. Faculty series 446
C. Lambert's series 448
Exercises on Chapter XII (174-199) 452
Chapter XIII. Divergent series. 457
59. General remarks on divergent series and the processes of limitation 457
60. The C- and H- processes 478
61. Application of C1- summation to the theory of Fourier series 492
62. The A- process 498
63. The E- process 507
Exercises on Chapter XIII (200-216) 516
Chapter XIV. Fuler's summation formula and asymptotic expansions. 618
64. Euler's summation formula 618
A. The summation formula 518
B. Applications 525
C. The evaluation of remiindcrs 531
65. Asymptotic series 535
66. Special cases of asymptotic expansions 543
A. Examples of the expansion problem 543
B. Examples of the summation problem 548
Exercises on Chapter XIV (217-225) 553
Bibliography 556
Name and subject index 557