Formal power series and generating functions. Operations with formal power series. Elementary generating functions Generating functions for well-known sequences Unambiguous formal grammars. The Lagrange theorem Analytic properties of functions represented as power series and their asymptotics of their coefficients Generating functions of several variables Partitions and decompositions Dirichlet generating functions and the inclusion-exclusion principle Enumeration of embedded graphs Final and bibliographical remarks Bibliography Index.

Lectures on Generating Functions

Shrinking the period lengths of continued fractions while still capturing convergents

We give a simple geometric proof that $e$ is irrational, using a construction of a nested sequence of closed intervals with intersection $e$. The proof leads to a new measure of irrationality for $e$: if $p$ and $q$ are integers with $q > 1$, then $|e - p/q| > 1/(S(q)+1)!$, where $S(q)$ is the smallest positive integer such that $S(q)!$ is a multiple of $q$. We relate this measure for $e$ to a known one and to the greatest prime factor of an integer. We make two conjectures and recall a theorem of Cantor that can be proved by a similar construction.

A geometric proof that $e$ is irrational and a new measure of its irrationality

/pdf/a-generalization-of-a-theorem-of-lekkerkerker-to-ostrowski-s-uwgbwaunr3.pdf

A generalization of a theorem of Lekkerkerker to Ostrowski's decomposition of natural numbers

The book is based on lectures given by the authors to undergraduate students at Moscow State University. It explains basic notions of “naive” set theory (cardinalities, ordered sets, transfinite induction, ordinals). The book can be read by undergraduate and graduate students and all those interested in basic notions of set theory. The book contains more than 100 problems of various degrees of difficulty. Library of Congress Cataloging-in-Publication Data Vereshchagin, Nikolai Konstantinovich, 1958– [Osnovy teorii mnozhestv. English] Basic set theory / A. Shen, N. K. Vereshchagin. p. cm. — (Student mathematical library, ISSN 1520-9121 ; v. 17) Authors’ names on t.p. of translation reversed from original. Includes bibliographical references and index. ISBN 0-8218-2731-6 (acid-free paper) 1. Set theory. I. Shen, A. (Alexander), 1958– II. Title. III. Series. QA248 .V4613 2002 511.3′22—dc21 2002066533 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 029042294, USA. Requests can also be made by e-mail to reprint-permission@ams.org. c © 2002 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ©∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 07 06 05 04 03 02

Basic set theory

Exploring the Number Jungle: A Journey into Diophantine Analysis@@@My Numbers, My Friends: Popular Lectures on Number Theory@@@The Prime Numbers and Their Distribution

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Exploring the Number Jungle: A Journey into Diophantine Analysis@@@My Numbers, My Friends: Popular Lectures on Number Theory@@@The Prime Numbers and Their Distribution