Cover 1
Title 6
Copyright 7
Contents 8
Preface 12
Notation and Terminology 14
Problems 16
Chapter 1. Real Numbers 18
1.1. Supremum and Infimum of Sets of Real Numbers. Continued Fractions 18
1.2. Some Elementary Inequalities 23
Chapter 2. Sequences of Real Numbers 34
2.1. Monotonic Sequences 34
2.2. Limits. Properties of Convergent Sequences 41
2.3. The Toeplitz Transformation, the Stolz Theorem and their Applications 50
2.4. Limit Points. Limit Superior and Limit Inferior 55
2.5. Miscellaneous Problems 62
Chapter 3. Series of Real Numbers 78
3.1. Summation of Series 78
3.2. Series of Nonnegative Terms 87
3.3. The Integral Test 103
3.4. Series of Positive and Negative Terms - Convergence, Absolute Convergence. Theorem of Leibniz 107
3.5. The Dirichlet and Abel Tests 114
3.6. Cauchy Product of Infinite Series 117
3.7. Rearrangement of Series. Double Series 120
3.8. Infinite Products 127
Solutions 138
Chapter 1. Real Numbers 140
1.1. Supremum and Infimum of Sets of Real Numbers. Continued Fractions 140
1.2. Some Elementary Inequalities 151
Chapter 2. Sequences of Real Numbers 166
2.1. Monotonic Sequences 166
2.2. Limits. Properties of Convergent Sequences 177
2.3. The Toeplitz Transformation, the Stolz Theorem and their Applications 196
2.4. Limit Points. Limit Superior and Limit Inferior 204
2.5. Miscellaneous Problems 223
Chapter 3. Series of Real Numbers 260
3.1. Summation of Series 260
3.2. Series of Nonnegative Terms 284
3.3. The Integral Test 317
3.4. Series of Positive and Negative Terms - Convergence, Absolute Convergence. Theorem of Leibniz 324
3.5. The Dirichlet and Abel Tests 339
3.6. Cauchy Product of Infinite Series 348
3.7. Rearrangement of Series. Double Series 357
3.8. Infinite Products 375
Bibliography - Books 394
Back Cover 396
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