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An Introduction to the Theory of Infinite Series
 
T. J. l’a. Bromwich
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
eBook ISBN:  978-1-4704-7336-5
Product Code:  CHEL/335.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $58.50
Add to cart
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An Introduction to the Theory of Infinite Series
T. J. l’a. Bromwich
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
eBook ISBN:  978-1-4704-7336-5
Product Code:  CHEL/335.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $58.50
Add to cart
  • Book Details
     
     
    AMS Chelsea Publishing
    Volume: 335; 1926; 535 pp
    MSC: Primary 40; Secondary 26; 30

    Third edition.
  • Table of Contents
     
     
    • Front Cover
    • PREFACE TO THE SECOND EDITION
    • PREFACE TO THE FIRST EDITION
    • CONTENTS
    • CHAPTER I. SEQUENCES AND LIMITS.
    • 1. Infinite sequences : convergence and divergence.
    • 2. Monotonic sequences; and conditions for their convergence.
    • 3. General principle of convergence.
    • 4. Solution of numerical equations by means of sequences.
    • 5·1. Upper and lower limits of a sequence.
    • 6. Sum of an infinite series ; addition of two series.
    • EXAMPLES.
    • CHAPTER II. SERIES OF POSITIVE TERMS.
    • 7.
    • 8. Comparison test for convergence (of positive series).
    • 9.
    • 10.
    • 11. Second test for convergence ; the logarithmic scale.
    • 12·1. Ratio-tests for convergence.
    • 13. Notes on the ratio-tests.
    • 14. Ermakoff's tests.
    • 15. Another sequence of tests.
    • 16. General notes on series of positive terms.
    • EXAMPLES.
    • CHAPTER III. SERIES IN GENERAL.
    • 17.
    • 18.
    • 19. Alternating serie
    • 20. Abel's Lemma.
    • 21.
    • 22.
    • 23.
    • 24. Transformation of slowly convergent alternating series.
    • EXAMPLES.
    • CHAPTER IV. ABSOLUTE CONVERGENCE.
    • 25.
    • 26.
    • 27. Applications of absolute convergence.
    • 28. Riemann's Theorem.
    • EXAMPLES.
    • CHAPTER V. DOUBLE SERIES.
    • 29.
    • 30. Repeated series.
    • 31. Double series of positive terms.
    • 32. Tests for convergence of a. double series of positive terms.
    • 33. Absolutely convergent double series.
    • 34.
    • 35.
    • 36. Substitution of a power-series in another power-series.
    • 37. Non-absolutely convergent double series.
    • EXAMPLES.
    • CHAPTER VI. INFINITE PRODUCTS.
    • 38. Weierstrass's Inequalities.
    • 39.
    • 40. Absolute convergence of an infinite product.
    • 41. Further tests for the convergence of infinite products in general.
    • 42. The Gamma-product.
    • EXAMPLES.
    • CHAPTER VII. SERIES OF VARIABLE TERMS.
    • 43. Uniform convergence of a sequence.
    • 44. Uniform convergence of a series.
    • 45. Fundamental properties of uniformly convergent series.
    • 46. Differentiation of an infinite series.
    • 47.
    • 48. Uniform convergence of an infinite product.
    • 49.
    • EXAMPLES.
    • CHAPTER VIII. POWER SERIES.
    • 50.
    • 51. Abel's theorem
    • 52. Properties of a power-series.
    • 53.
    • 54. Multiplication and division of power-series.
    • 55. Reversion of a power-series.
    • 56. Applications to the theory of differential equations.
    • 57. The exponential limit.
    • 58. The exponential function.
    • 59. The sine and cosine power-series.
    • 60. Other methods of establishing the sine and cosine power-series.
    • 61. The genera.I binomial theorem.
    • 62. The logarithmic series.
    • 63.
    • 64. The power-series for arc sin x and arc tan x.
    • 65. Various trigonometrical power-series.
    • EXAMPLES A
    • EXAMPLES B.
    • CHAPTER IX. TRIGONOMETRICAL FORMULAE.
    • 66. Expressions for cos nΘ and (sin nΘ/sin Θ) as polynomials in cos Θ.
    • 67. Forms for cos nΘ and sin nΘ in terms of sin Θ.
    • 68.
    • 69. Various deductions from Art. 67.
    • 70. Expressions of sinΘ and cos Θ as infinite products.
    • 71. Weierstrass's formula for the sine-product.
    • EXAMPLES.
    • CHAPTER X. COMPLEX SERIES AND PRODUCTS.
    • 72. The algebra of complex numbers.
    • 73. Argand's diagram.
    • 74. Multiplication; de Moivre's theorem.
    • 75. General principle of convergence for complex. sequences.
    • 76. Absolute convergence of a series of complex terms.
    • 77. Extension of Maclaurin's integral test.
    • 78. Ratio-Tests for absolute convergence.
    • 79. Ratio-tests for non-absolute convergence.
    • 80. Abel's Lemma (for complex series).
    • 81. Further tests for convergence.
    • 82. Double series of complex terms.
    • 83. Uniform convergence.
    • 84. Circle of convergence of a power-series Σanxn.
    • 85. Behaviour of a power-series on the circle of convergence.
    • 86. Abel's theorem and allied theorems.
    • 87. Poisson's integral.
    • 88. Taylor's theorem for a power-series.
    • 89. Extensions of Cauchy's inequalities.
    • 90. Lagrange's series.
    • 91. Weierstrass's double-series theorem.
    • EXAMPLES.
    • CHAPTER XI. SPECIAL COMPLEX SERIES AND FUNCTIONS.
    • 92. The exponential power-series.
    • 93. Connexion between the exponential and circular functions.
    • 94. The logarithm and its principal branch.
    • 95. The logarithmic power-series.
    • 96. The binomial power-series.
    • 97. The remainder in the binomial series.
    • 98. The infinite products for sin x and cos x.
    • 99. The series of fractions for cot x, tan x, cosec x.
    • 100. The power-series for x/ ( ex-1 ).
    • 101. Bernoullian functions.
    • 102. Euler's summation formula.
    • 103. Development of elliptic function formulae from the algebraic side.
    • EXAMPLES.
    • CHAPTER XII. ASYMPTOTIC SERIES AND TRIGONOMETRICAL SERIES.
    • 104. Historical remarks on the use of non-convergent series.
    • 105. General considerations on non-convergent series.
    • 106. Euler's use of asymptotic series.
    • 107. The remainder in Euler's formula.
    • 108. Application of Euler's formula to Stirling's series.
    • 109. Calculation of integrals by means of asymptotic series.
    • 110. Asymptotic series for integrals containing sines and cosines.
    • 111. Stirling's series.
    • 112. Stokes' asymptotic expression for the series
    • 113. Poincare's theory of asymptotic series.
    • 114. Applications of Poincare's theory.
    • 115. Differential equations.
    • 116. The modified Bessel's equation.
    • 117. Identification of the solutions of Art. 116 with known solutions.
    • 118. The ordinary Bessel function.
    • 119.
    • 120. Topics included in the present section.
    • 121. Series which can be summed directly.
    • 122. Series which can be summed by integration.
    • 123. Recognition of discontinuities in the sum of a trigonometrical series.
    • 124. Differentiation of trigonometrical series.
    • 125. Extension of the method of Art. 124.
    • 126. Dirichlet's summation of Fourier's series.
    • 127. Summation of sine- and cosine-series.
    • 128. Stokes's transformation for finding discontinuities and for differentiating a Fourier series.
    • 129. Fejer's theorem on Fourier's series.
    • 130. Poisson's Integral.
    • 131. Character of the approximation curves near a discontinuity in a Fourier-series.
    • 132. Fejer's lemma.
    • EXAMPLES A.
    • EXAMPLES B.
    • APPENDIX I. ARITHMETIC THEORY OF IRRATIONAL NUMBERS AND LIMITS.
    • 133. Infinite decimals.
    • 134. The order of the system of infinite decimals.
    • 135. Additional arithmetical examples of infinite decimals which are not rational.
    • 136. Geometrical examples.
    • 137. A special classification of rational numbers.
    • 138. Dedekind's definition of irrational numbers.
    • 139. Definitions of equal, greater, less ; deductions.
    • 140. Deductions from the definitions.
    • 141. Modified form of Dedekind's definition.
    • 142. Algebraic operations with irrational numbers.
    • 143. The principle of convergence for monotonic sequences whose terms may be either rational or irrational.
    • 144. Maximum and minimum limiting values of a sequence of rational or irrational terms.
    • 145. The general principle of convergence is both necessary and sufficient.
    • 146. First theorem on limits of quotients.
    • 147. Second theorem on limits of quotients.
    • 148. An extension of Abel's Lemma.
    • 149. Cauchy's theorems.
    • 150. Cesaro's theorem.
    • 151. The Hardy-Landau converse of Cauchy's first theorem.
    • 152.
    • EXAMPLES.
    • APPENDIX II. DEFINITIONS OF THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS.
    • 153.
    • 154. Definition of the logarithmic function.
    • 155. Fundamental properties of the logarithmic function.
    • 156. Napier's Logarithms.
    • 157. Napier's method of calculating logarithms.
    • 158. The exponential function.
    • 159. Some miscellaneous inequalities.
    • 160. Some limits ; the logarithmic scale of infinity.
    • 161. The existence of an area for the rectangular hyperbola.
    • 162. Extension of the discussion to curves which are not monotonic.
    • 163. Extension of definition to double integrals.
    • EXAMPLES.
    • APPENDIX III. SOME THEOREMS ON INFINITE INTEGRALS AND GAMMA-FUNCTIONS.
    • 164. Infinite integrals : definitions.
    • 165. Special case of monotonic functions.
    • 166. Tests of convergence for infinite integrals with a positive integrand.
    • 167. Examples.
    • 168. Analogue of Abel's Lemma..
    • 169. Tests of convergence in general.
    • 170. Frullani's integrals.
    • 171. Uniform convergence of an infinite integral.
    • 172. Applications of uniform convergence.
    • 173. Applications of Art. 172.
    • 174. Some further theorems on integrals containing another variable.
    • 175. Integration of series, when infinities of the integrand occur in the range.
    • 176. Integration of an infinite series over an infinite interval.
    • 177. The inversion of a repeated inflnite integral.
    • 178. The Gamma-integral.
    • 179. Stirling's asymptotic formula for the Gamma-function when x is real, large and positive.
    • 180. Integrals for log Γ(l +x).
    • EXAMPLES.
    • MISCELLANEOUS EXAMPLES.
    • INDEX OF SPECIAL INTEGRALS, PRODUCTS, AND SERIES.
    • GENERAL INDEX.
    • Back Cover
  • Additional Material
     
     
  • Reviews
     
     

    • The book is especially good at counterexamples, and includes many of these to warn against pitfalls in reasoning and to show that all the hypotheses of the theorems are really needed. One especially nice feature is the use of Tannery's theorem, on interchanging limit and summation, throughout the book.
      MAA Reviews
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
AMS Chelsea Publishing
Volume: 335; 1926; 535 pp
MSC: Primary 40; Secondary 26; 30

Third edition.
  • Front Cover
  • PREFACE TO THE SECOND EDITION
  • PREFACE TO THE FIRST EDITION
  • CONTENTS
  • CHAPTER I. SEQUENCES AND LIMITS.
  • 1. Infinite sequences : convergence and divergence.
  • 2. Monotonic sequences; and conditions for their convergence.
  • 3. General principle of convergence.
  • 4. Solution of numerical equations by means of sequences.
  • 5·1. Upper and lower limits of a sequence.
  • 6. Sum of an infinite series ; addition of two series.
  • EXAMPLES.
  • CHAPTER II. SERIES OF POSITIVE TERMS.
  • 7.
  • 8. Comparison test for convergence (of positive series).
  • 9.
  • 10.
  • 11. Second test for convergence ; the logarithmic scale.
  • 12·1. Ratio-tests for convergence.
  • 13. Notes on the ratio-tests.
  • 14. Ermakoff's tests.
  • 15. Another sequence of tests.
  • 16. General notes on series of positive terms.
  • EXAMPLES.
  • CHAPTER III. SERIES IN GENERAL.
  • 17.
  • 18.
  • 19. Alternating serie
  • 20. Abel's Lemma.
  • 21.
  • 22.
  • 23.
  • 24. Transformation of slowly convergent alternating series.
  • EXAMPLES.
  • CHAPTER IV. ABSOLUTE CONVERGENCE.
  • 25.
  • 26.
  • 27. Applications of absolute convergence.
  • 28. Riemann's Theorem.
  • EXAMPLES.
  • CHAPTER V. DOUBLE SERIES.
  • 29.
  • 30. Repeated series.
  • 31. Double series of positive terms.
  • 32. Tests for convergence of a. double series of positive terms.
  • 33. Absolutely convergent double series.
  • 34.
  • 35.
  • 36. Substitution of a power-series in another power-series.
  • 37. Non-absolutely convergent double series.
  • EXAMPLES.
  • CHAPTER VI. INFINITE PRODUCTS.
  • 38. Weierstrass's Inequalities.
  • 39.
  • 40. Absolute convergence of an infinite product.
  • 41. Further tests for the convergence of infinite products in general.
  • 42. The Gamma-product.
  • EXAMPLES.
  • CHAPTER VII. SERIES OF VARIABLE TERMS.
  • 43. Uniform convergence of a sequence.
  • 44. Uniform convergence of a series.
  • 45. Fundamental properties of uniformly convergent series.
  • 46. Differentiation of an infinite series.
  • 47.
  • 48. Uniform convergence of an infinite product.
  • 49.
  • EXAMPLES.
  • CHAPTER VIII. POWER SERIES.
  • 50.
  • 51. Abel's theorem
  • 52. Properties of a power-series.
  • 53.
  • 54. Multiplication and division of power-series.
  • 55. Reversion of a power-series.
  • 56. Applications to the theory of differential equations.
  • 57. The exponential limit.
  • 58. The exponential function.
  • 59. The sine and cosine power-series.
  • 60. Other methods of establishing the sine and cosine power-series.
  • 61. The genera.I binomial theorem.
  • 62. The logarithmic series.
  • 63.
  • 64. The power-series for arc sin x and arc tan x.
  • 65. Various trigonometrical power-series.
  • EXAMPLES A
  • EXAMPLES B.
  • CHAPTER IX. TRIGONOMETRICAL FORMULAE.
  • 66. Expressions for cos nΘ and (sin nΘ/sin Θ) as polynomials in cos Θ.
  • 67. Forms for cos nΘ and sin nΘ in terms of sin Θ.
  • 68.
  • 69. Various deductions from Art. 67.
  • 70. Expressions of sinΘ and cos Θ as infinite products.
  • 71. Weierstrass's formula for the sine-product.
  • EXAMPLES.
  • CHAPTER X. COMPLEX SERIES AND PRODUCTS.
  • 72. The algebra of complex numbers.
  • 73. Argand's diagram.
  • 74. Multiplication; de Moivre's theorem.
  • 75. General principle of convergence for complex. sequences.
  • 76. Absolute convergence of a series of complex terms.
  • 77. Extension of Maclaurin's integral test.
  • 78. Ratio-Tests for absolute convergence.
  • 79. Ratio-tests for non-absolute convergence.
  • 80. Abel's Lemma (for complex series).
  • 81. Further tests for convergence.
  • 82. Double series of complex terms.
  • 83. Uniform convergence.
  • 84. Circle of convergence of a power-series Σanxn.
  • 85. Behaviour of a power-series on the circle of convergence.
  • 86. Abel's theorem and allied theorems.
  • 87. Poisson's integral.
  • 88. Taylor's theorem for a power-series.
  • 89. Extensions of Cauchy's inequalities.
  • 90. Lagrange's series.
  • 91. Weierstrass's double-series theorem.
  • EXAMPLES.
  • CHAPTER XI. SPECIAL COMPLEX SERIES AND FUNCTIONS.
  • 92. The exponential power-series.
  • 93. Connexion between the exponential and circular functions.
  • 94. The logarithm and its principal branch.
  • 95. The logarithmic power-series.
  • 96. The binomial power-series.
  • 97. The remainder in the binomial series.
  • 98. The infinite products for sin x and cos x.
  • 99. The series of fractions for cot x, tan x, cosec x.
  • 100. The power-series for x/ ( ex-1 ).
  • 101. Bernoullian functions.
  • 102. Euler's summation formula.
  • 103. Development of elliptic function formulae from the algebraic side.
  • EXAMPLES.
  • CHAPTER XII. ASYMPTOTIC SERIES AND TRIGONOMETRICAL SERIES.
  • 104. Historical remarks on the use of non-convergent series.
  • 105. General considerations on non-convergent series.
  • 106. Euler's use of asymptotic series.
  • 107. The remainder in Euler's formula.
  • 108. Application of Euler's formula to Stirling's series.
  • 109. Calculation of integrals by means of asymptotic series.
  • 110. Asymptotic series for integrals containing sines and cosines.
  • 111. Stirling's series.
  • 112. Stokes' asymptotic expression for the series
  • 113. Poincare's theory of asymptotic series.
  • 114. Applications of Poincare's theory.
  • 115. Differential equations.
  • 116. The modified Bessel's equation.
  • 117. Identification of the solutions of Art. 116 with known solutions.
  • 118. The ordinary Bessel function.
  • 119.
  • 120. Topics included in the present section.
  • 121. Series which can be summed directly.
  • 122. Series which can be summed by integration.
  • 123. Recognition of discontinuities in the sum of a trigonometrical series.
  • 124. Differentiation of trigonometrical series.
  • 125. Extension of the method of Art. 124.
  • 126. Dirichlet's summation of Fourier's series.
  • 127. Summation of sine- and cosine-series.
  • 128. Stokes's transformation for finding discontinuities and for differentiating a Fourier series.
  • 129. Fejer's theorem on Fourier's series.
  • 130. Poisson's Integral.
  • 131. Character of the approximation curves near a discontinuity in a Fourier-series.
  • 132. Fejer's lemma.
  • EXAMPLES A.
  • EXAMPLES B.
  • APPENDIX I. ARITHMETIC THEORY OF IRRATIONAL NUMBERS AND LIMITS.
  • 133. Infinite decimals.
  • 134. The order of the system of infinite decimals.
  • 135. Additional arithmetical examples of infinite decimals which are not rational.
  • 136. Geometrical examples.
  • 137. A special classification of rational numbers.
  • 138. Dedekind's definition of irrational numbers.
  • 139. Definitions of equal, greater, less ; deductions.
  • 140. Deductions from the definitions.
  • 141. Modified form of Dedekind's definition.
  • 142. Algebraic operations with irrational numbers.
  • 143. The principle of convergence for monotonic sequences whose terms may be either rational or irrational.
  • 144. Maximum and minimum limiting values of a sequence of rational or irrational terms.
  • 145. The general principle of convergence is both necessary and sufficient.
  • 146. First theorem on limits of quotients.
  • 147. Second theorem on limits of quotients.
  • 148. An extension of Abel's Lemma.
  • 149. Cauchy's theorems.
  • 150. Cesaro's theorem.
  • 151. The Hardy-Landau converse of Cauchy's first theorem.
  • 152.
  • EXAMPLES.
  • APPENDIX II. DEFINITIONS OF THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS.
  • 153.
  • 154. Definition of the logarithmic function.
  • 155. Fundamental properties of the logarithmic function.
  • 156. Napier's Logarithms.
  • 157. Napier's method of calculating logarithms.
  • 158. The exponential function.
  • 159. Some miscellaneous inequalities.
  • 160. Some limits ; the logarithmic scale of infinity.
  • 161. The existence of an area for the rectangular hyperbola.
  • 162. Extension of the discussion to curves which are not monotonic.
  • 163. Extension of definition to double integrals.
  • EXAMPLES.
  • APPENDIX III. SOME THEOREMS ON INFINITE INTEGRALS AND GAMMA-FUNCTIONS.
  • 164. Infinite integrals : definitions.
  • 165. Special case of monotonic functions.
  • 166. Tests of convergence for infinite integrals with a positive integrand.
  • 167. Examples.
  • 168. Analogue of Abel's Lemma..
  • 169. Tests of convergence in general.
  • 170. Frullani's integrals.
  • 171. Uniform convergence of an infinite integral.
  • 172. Applications of uniform convergence.
  • 173. Applications of Art. 172.
  • 174. Some further theorems on integrals containing another variable.
  • 175. Integration of series, when infinities of the integrand occur in the range.
  • 176. Integration of an infinite series over an infinite interval.
  • 177. The inversion of a repeated inflnite integral.
  • 178. The Gamma-integral.
  • 179. Stirling's asymptotic formula for the Gamma-function when x is real, large and positive.
  • 180. Integrals for log Γ(l +x).
  • EXAMPLES.
  • MISCELLANEOUS EXAMPLES.
  • INDEX OF SPECIAL INTEGRALS, PRODUCTS, AND SERIES.
  • GENERAL INDEX.
  • Back Cover

  • The book is especially good at counterexamples, and includes many of these to warn against pitfalls in reasoning and to show that all the hypotheses of the theorems are really needed. One especially nice feature is the use of Tannery's theorem, on interchanging limit and summation, throughout the book.
    MAA Reviews
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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