## A First Course in Mathematical AnalysisThis straightforward course based on the idea of a limit is intended for students who have acquired a working knowledge of the calculus and are ready for a more systematic treatment which also brings in other limiting processes, such as the summation of infinite series and the expansion of trigonometric functions as power series. Particular attention is given to clarity of exposition and the logical development of the subject matter. A large number of examples is included, with hints for the solution of many of them. |

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### Contents

II | 1 |

III | 2 |

IV | 3 |

V | 7 |

VI | 8 |

VII | 10 |

VIII | 13 |

IX | 15 |

XLIII | 92 |

XLIV | 94 |

XLV | 96 |

XLVI | 97 |

XLVII | 99 |

XLVIII | 101 |

XLIX | 104 |

L | 105 |

X | 18 |

XI | 19 |

XII | 23 |

XIII | 25 |

XIV | 26 |

XV | 28 |

XVI | 31 |

XVII | 32 |

XVIII | 34 |

XIX | 38 |

XX | 39 |

XXI | 40 |

XXII | 43 |

XXIII | 47 |

XXIV | 49 |

XXVI | 51 |

XXVII | 53 |

XXVIII | 56 |

XXIX | 57 |

XXX | 60 |

XXXI | 62 |

XXXII | 65 |

XXXIII | 67 |

XXXIV | 69 |

XXXV | 72 |

XXXVI | 73 |

XXXVII | 75 |

XXXVIII | 77 |

XXXIX | 78 |

XL | 82 |

XLI | 88 |

XLII | 90 |

### Common terms and phrases

absolutely convergent algebra analysis approximation axioms Calculate circle closed interval coefficients complex numbers constant continuous function converges to sum Corollary corresponding cosx curve decreasing sequence Deduce defined definition derivative differentiable dissection diverges equation example exercises are given exists expx finite number formula geometric series gives greater illustration inequality infimum infinite series integral inverse function irrational less maxima and minima maximum mean value theorem method negative notation Notes nth term null sequence open interval oscillates finitely polynomial positive integer positive number positive terms power series Proof properties Prove radius of convergence rational function rational numbers reader real numbers right-hand side Rolle's theorem roots satisfies sequence sn series converges set of numbers sinx sn oscillates strictly increasing subinterval Suppose supremum tangent Taylor's theorem theorem 6.2 triangle trigonometric functions upper bound variable write zero