## Excursions in Calculus: An Interplay of the Continuous and the Discrete, Volume 13The purpose of this book is to explore the rich and elegant interplay that exists between the two main currents of mathematics, the continuous and the discrete. Such fundamental notions in discrete mathematics as induction, recursion, combinatorics, number theory, discrete probability, and the algorithmic point of view as a unifying principle are continually explored as they interact with traditional calculus. The book is addressed primarily to well-trained calculus students and those who teach them, but it can also serve as a supplement in a traditional calculus course for anyone who wants to see more. The problems, taken for the most part from probability, analysis, and number theory, are an integral part of the text. There are over 400 problems presented in this book. |

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### Contents

Function and Form | 123 |

On the Average | 185 |

From Pi to the Prime Number Theorem | 247 |

A Potpourri | 309 |

The Congruence Notation | 369 |

379 | |

Sources for Solutions | 397 |

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### Common terms and phrases

algebra algorithm American Mathematical Monthly approximation arithmetic mean asserts Bernoulli numbers binomial calculus Cantor set century circle coefficients congruence conjecture consecutive continuous function converges curve defined denote derived digits discovered disks divisible equal equation error Euclidean algorithm Euler example Fermat's last theorem Fermat's little theorem Fibonacci numbers formula fraction Gauss geometric mean given greatest common divisor Hint inequality interval iteration known Leibniz logarithmic logarithmic spiral Mandelbrot math mathematical induction mathematicians method multiple natural number number of primes number theory Numbers 385 Observe obtain Pascal Pascal's triangle pattern perfect number polynomial positive integers positive numbers prime numbers probability proof Prove Pythagorean triangle Pythagorean triple rational numbers real number rectangle recursive relatively prime remarkable result sequence Show sides simple solution square theorem Theory of Numbers triangular numbers Unsolved values zero