## Mathematical Analysis and Applications: An IntroductionMathematical Analysis and Applications is a first introduction to higher mathematics for students. Starting with a discussion of real numbers and functions, the text introduces standard topics of differential and integral calculus together with their applications such as differential equations, numerical analysis, and approximation methods.The subject is developed in an integrated manner, bringing out its essential unity and the interrelationship between these topics. The text is written in an informal manner and in lively language, but without sacrificing rigour in thinking and precision in writing, so essential in mathematics.The book is divided in four Parts and seventeen Chapters, described below. A brief Historical Note is included at the end of each Part that describes the contributions of the mathematicians mentioned in the Text. The Book is ideally suited for class-room teaching and also for self-study to undergraduate students of Mathematics and of other disciplines such as Statistics, Physics, Computer Sciences or Engineering that use Mathematical Methods extensively. It is also a handy reference to teachers of Mathematics. |

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### Contents

RATIONAL NUMBERS | 5 |

INDUCTION PRINCIPLE | 19 |

REAL NUMBER SYSTEM | 33 |

SETS AND FUNCTIONS | 49 |

SEQUENCES | 69 |

CONTINUOUS FUNCTIONS I | 89 |

CONTINUOUS FUNCTIONS II | 107 |

INFINITE SERIES | 131 |

MEAN VALUE THEOREMS | 171 |

TAYLORS THEOREM APPROXIMATION | 189 |

POWER SERIES MAPS | 205 |

RIEMANN INTEGRATION I | 277 |

RIEMANN INTEGRATION II | 301 |

RIEMANN INTEGRATION III | 313 |

INDEX | 338 |

DIFFERENTIABLE FUNCTIONS | 157 |

### Common terms and phrases

addition answer apply approximation axiom bounded called Chapter choose clear closed compact Consider constant continuous function converges defined definition denoted derivative differentiable discussion domain earlier equal equation evaluate example Exercises exists expression f is continuous fact finite fixed formula function f further given gives graph Hence holds increasing infinite integral interval inverse irrational keep least limit point linear Mathematics Mean Value Theorem means method mind namely natural notation Note obtained particular solution polynomial power series problems PROOF properties Proposition proved radius of convergence rational real numbers respect result Riemann integral rule satisfies sequence side Similarly simple solution solving statement subset Suppose Taylor's Theorem true variables write zero