Theory and applications of infinite series

This unusually clear and interesting classic offers a thorough and reliable treatment of an important branch of higher analysis. The work 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, and other topics). Exercises throughout. Ideal for self-study. Cover......Page 1 Publisher......Page 2 Title page......Page 3 Copyright page......Page 4 From the preface to the first (German) edition......Page 5 Preface to the fourth (German) edition......Page 7 Preface to the second English edition......Page 8 Contents......Page 9 Introduction......Page 13 § 1. The system of rational numbers and its gaps......Page 15 § 2. Sequences of rational numbers......Page 26 § 3. Irrational numbers......Page 35 § 4. Completeness and uniqueness of the system of real numbers......Page 45 § 5. Radix fractions and the Dedekind section......Page 49 Exercises on Chapter I (1—8)......Page 54 § 6. Arbitrary sequences and arbitrary null sequences......Page 55 § 7. Powers, roots, and logarithms Special null sequences......Page 61 § 8. Convergent sequences......Page 76 § 9. The two main criteria......Page 90 § 10. Limiting points and upper and lower limits......Page 101 § 11. Infinite series, infinite products, and infinite continued fractions......Page 110 Exercises on Chapter II (9—33)......Page 118 § 12. The first principal criterion and the two comparison tests......Page 122 § 13. The root test and the ratio test......Page 128 § 14. Series of positive, monotone decreasing terms......Page 132 Exercises on Chapter III (34—44)......Page 137 § 15. The second principal criterion and the algebra of convergent series......Page 138 § 16. Absolute convergence. Derangement of series......Page 148 § 17. Multiplication of infinite series......Page 158 Exercises on Chapter IV (45—03)......Page 161 § 18. The radius of convergence......Page 163 § 19. Functions of a real variable......Page 170 § 20. Principal properties of functions represented by power series......Page 183 § 21. The algebra of power series......Page 191 Exercises on Chapter V (64-73)......Page 200 § 22. The rational functions......Page 201 § 23. The exponential function......Page 203 § 24. The trigonometrical functions......Page 210 § 25. The binomial series......Page 220 § 26. The logarithmic series......Page 223 § 27. The cyclometrical functions......Page 225 Exercises on Chapter VI (74-84)......Page 227 § 28. Products with positive terms......Page 230 § 29. Products with arbitrary terms. Absolute convergence......Page 233 § 30. Connection between series and products. Conditional and unconditional convergence......Page 238 Exercises on Chapter VII (85—99)......Page 240 § 31. Statement of the problem......Page 242 § 32. Evaluation of the sum of a series by means of a closed expression......Page 244 § 33. Transformation of series......Page 252 § 34. Numerical evaluations......Page 259 § 35. Applications of the transformation of series to numerical evaluations......Page 272 Exercises on Chapter VIII (100—132)......Page 279 § 36. Detailed study of the two comparison tests......Page 286 § 37. The logarithmic scales......Page 290 § 38. Special comparison tests of the second kind......Page 296 § 39. Theorems of Abel, Dini, and Pringsheim, and their application to a fresh deduction of the logarithmic scale of comparison tests......Page 302 § 40. Series of monotonely diminishing positive terms......Page 306 § 41. General remarks on the theory of the convergence and divergence of series of positive terms......Page 310 § 42. Systematization of the general theory of convergence......Page 317 Exercises on Chapter IX (138—141)......Page 323 § 43. Tests of convergence for series ot arbitrary terms......Page 324 § 44. Rearrangement of conditionally convergent series......Page 330 § 45. Multiplication of conditionally convergent series......Page 332 Exercises on Chapter X (142—153)......Page 336 § 46. Uniform convergence......Page 338 § 47. Passage to the limit term by term......Page 350 § 48. Tests of uniform convergence......Page 356 A. Euler's formulae......Page 362 B. Dirichlet's integral......Page 368 C. Conditions of convergence......Page 376 § 50. Applications of the theory of Fourier series......Page 384 § 51. Products with variable terms......Page 392 Exercises on Chapter XI (154—173)......Page 397 § 52. Complex numbers and sequences......Page 400 § 53. Series of complex terms......Page 408 § 54. Power series. Analytic functions......Page 413 I. Rational functions......Page 422 II. The exponential function......Page 423 III. The functions $\cos z$ and $\sin z$......Page 426 IV. The functions $\cot z$ and $\tan z$......Page 429 V. The logarithmic series......Page 431 VI. The inverse sine series......Page 433 VII. The inverse tangent series......Page 434 VIII. The binomial series......Page 435 § 56. Series of variable terms. Uniform convergence. Weierstrass' theorem on double series......Page 440 § 57. Products with complex terms......Page 446 A. Dirichlet's series......Page 453 B. Faculty series......Page 458 C. Lambert's series......Page 460 Exercises on Chapter XII (174—199)......Page 464 § 59. General remarks on divergent series and the processes of limitation......Page 469 § 60. The $C$- and $H$- processes......Page 490 § 61. Application of $C_1$- summation to the theory of Fourier series......Page 504 § 62. The $A$- process......Page 510 § 63. The $E$- process......Page 519 Exercises on Chapter XIII (200—216)......Page 528 A. The summation formula......Page 530 B. Applications......Page 537 C. The evaluation of remainders......Page 543 § 65. Asymptotic series......Page 547 A. Examples of the expansion problem......Page 555 B. Examples of the summation problem......Page 560 Exercises on Chapter XIV (217-225)......Page 565 Bibliography......Page 568 Name and subject index......Page 569 This classic work, written in a clear and interesting style, with many exercises, offers a thorough and reliable treatment of an important branch of higher analysis. It lends itself well to use in course work; however, because of its consistent clear illustrations of theoretical difficulties, the book is also ideal for self-study.Since all higher analysis depends on the theory of numbers, Professor Knopp (formerly Professor of Mathematics, University of Tübingen) begins with an introduction to the theory of real numbers, an indispensable foundation for what is to come. This introduction is followed by an extensive account of the theory of sequences and the actual theory of infinite series. The latter is covered in two stages: (1) the classical theory (2) later developments of the 19th century.Carefully selected exercises have been included throughout, emphasizing applications of the theory, rather than purely theoretical considerations.Aimed at students already acquainted with the elements of differential and integral calculus, this work grew out of the author's lectures and course work at the universities of Berlin and Königsberg. This pedagogical background helped him achieve a work of utmost clarity and precision — one that belongs in the library of every serious mathematician or student of higher analysis.