problems and solutions in real analysis
TRANSCRIPT
PROBLEMS AND SOLUTIONS IN REAL ANALYSIS
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Masayoshi Hata Kyoto University, Japan
Y||* World Scientific N E W J E R S E Y • L O N D O N • S 1 N G A P 0 R E • B E I J I N G • S H A N G H A I • H O N G K O N G • T A I P E I •
Contents
Preface v
1. Sequences and Limits 1
Solutions 5
2. Infinite Series 15
Solutions 20
3. Continuous Functions 31
Solutions 35
4. Differentiation 43
Solutions 49
5. Integration 59
Solutions 66
6. Improper Integrals 77
Solutions 81
7. Series of Functions 93
Solutions 100
8. Approximation by Polynomials 113
Solutions 117
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X Problems and Solutions in Real Analysis
9. Convex Functions 125
Solutions 129
10. Various proofs of £(2) = n2/6 139
Solutions 146
11. Functions of Several Variables 157
Solutions 161
12. Uniform Distribution 171
Solutions 174
13. Rademacher Functions 181
Solutions 185
14. Legendre Polynomials 191
Solutions 195
15. Chebyshev Polynomials 205
Solutions 209
16. Gamma Function 219
Solutions 225
17. Prime Number Theorem 239
Solutions 245
18. Miscellanies 257
Solutions 263
Bibliography 273
Index 285