| Foreword |
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vii | |
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1 The Greek era: Speculation, ideas and the comet of rigor |
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1 | (10) |
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1.1 Infinity, challenging Greek philosophers |
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1 | (1) |
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1.2 Infinity, mastered by a Greek "geometer": Eudoxus |
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2 | (3) |
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1.3 Archimedes and the geometric series to be |
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5 | (4) |
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1.4 Memorizing a joint venture |
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9 | (2) |
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2 Successful experiments on infinite summation |
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11 | (8) |
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2.1 Oresmus and the geometric series |
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11 | (2) |
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2.2 Freestyle: Cavalieri summing indivisibles |
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13 | (3) |
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16 | (3) |
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3 Series of functions: the prototypes |
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19 | (48) |
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19 | (4) |
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3.1.1 Series versus sequences |
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19 | (1) |
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3.1.2 Series: evaluation and expansion |
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20 | (3) |
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3.2 The power of power series |
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23 | (23) |
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3.2.1 Nilakantha and the arcustangent series |
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23 | (3) |
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3.2.2 The hyperbolic logarithm and its series |
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26 | (3) |
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3.2.3 Newton and the binomial series |
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29 | (3) |
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3.2.4 Taylor series and expansion; ordinary power series |
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32 | (4) |
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3.2.5 The binomial series in the hands of Euler |
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36 | (6) |
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3.2.6 Euler's algebraical analysis: evaluation and valuation |
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42 | (4) |
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3.3 The power of trigonometric series |
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46 | (9) |
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46 | (1) |
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3.3.2 Euler evaluating a trigonometric series |
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46 | (3) |
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49 | (1) |
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3.3.4 The mysterious ... Fourier series |
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50 | (5) |
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55 | (12) |
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3.4.1 Series of powers in the hands of Cauchy and Laurent |
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55 | (6) |
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61 | (6) |
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4 Series seriously: the Greek comet reappears |
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67 | (22) |
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4.1 Rigor in retrospect and prospect |
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67 | (2) |
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4.2 Farewell to the infinitely small; the "epsilontics" |
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69 | (9) |
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4.2.1 Convergence, continuity |
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69 | (2) |
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4.2.2 Dirichlet, Heine: uniform continuity |
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71 | (1) |
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4.2.3 From Abel to Weierstraß: uniform convergence |
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72 | (6) |
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4.3 Welcome to irrationals: the complete space of real numbers |
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78 | (6) |
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4.4 Another complete space: the home of convergent sequences |
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84 | (5) |
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5 On the verge of summability |
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89 | (22) |
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5.1 Divergent series: suspected and respected |
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89 | (1) |
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5.2 The initiation of Cesaro and Abel summation |
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90 | (8) |
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5.2.1 Grandi's series, recurrent series |
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90 | (2) |
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5.2.2 Frobenius' theorem and the limit theorems of Cauchy and Abel |
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92 | (2) |
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5.2.3 The Cauchy product revisited by Abel and Cesaro |
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94 | (1) |
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5.2.4 Introducing the methods of basic Cesaro and of Abel means |
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95 | (3) |
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5.3 Features of Cesaro and Abel means |
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98 | (13) |
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5.3.1 Inclusion, limitation, efficiency |
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98 | (3) |
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5.3.2 The C1 limit: a continuous functional on a Banach space |
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101 | (2) |
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5.3.3 An early triumph of arithmetic means: Fejer's Theorem |
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103 | (1) |
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5.3.4 On the scale of Cesaro means |
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103 | (3) |
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5.3.5 Cesaro's Cauchy product |
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106 | (1) |
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5.3.6 Inverse theorems, Tauber's theorem |
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107 | (4) |
| Appendixes (of notes, proofs, exercises) |
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111 | (8) |
| Life Data |
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119 | (2) |
| References |
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121 | (6) |
| Index |
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127 | |
Lisainfo e-raamatute kohta