MATHEMATICAL ANALYSIS

By X. BEERT

CONTENTS NUMBERS REAL NUMBERS INEQUALITIES PROVING INEQUALITIES LEAST UPPER BOUNDS AND GREATEST LOWER BOUNDS MANIPULATING REAL NUMBERS EXERCISES SEQUENCES INTRODUCING SEQUENCES NULL SEQUENCES CONVERGENT SEQUENCES DIVERGENT SEQUENCES THE MONOTONE CONVERGENCE THEOREM SERIES INTRODUCING SERIES SERIES WITH NON-NEGATIVE TERMS SERIES WITH POSITIVE AND NEGATIVE TERMS THE EXPONENTIAL FUNCTION X 7! EX CONTINUITY CONTINUOUS FUNCTIONS PROPERTIES OF CONTINUOUS FUNCTIONS INVERSE FUNCTIONS DEFINING EXPONENTIAL FUNCTIONS LIMITS AND CONTINUITY LIMITS OF FUNCTIONS ASYMPTOTIC BEHAVIOUR OF FUNCTIONS LIMITS OF FUNCTIONS – USING " AND CONTINUITY – USING " AND UNIFORM CONTINUITY DIFFERENTIATION DIFFERENTIABLE FUNCTIONS RULES FOR DIFFERENTIATION ROLLE’S THEOREM THE MEAN VALUE THEOREM L’HOˆPITAL’S RULE THE BLANCMANGE FUNCTION INTEGRATION THE RIEMANN INTEGRAL PROPERTIES OF INTEGRALS FUNDAMENTAL THEOREM OF CALCULUS INEQUALITIES FOR INTEGRALS AND THEIR APPLICATIONS STIRLING’S FORMULA FOR N! POWER SERIES TAYLOR POLYNOMIALS TAYLOR’S THEOREM CONVERGENCE OF POWER SERIES MANIPULATING POWER SERIES NUMERICAL ESTIMATES FOR P