þ 1978 1977 ¼ 1978 1: The answer was obtained! After winning the City Olympiad, I was sent to the Regional Math Olympiad and was again surprised that two or three problems there were on topics that were not yet covered in our classes at school. Then I applied the formula of the difference of two squares, which made the denominator of the fraction one. I multiplied the numerator and denominator of one fraction by the quantity that differed from its denominator by only the sign. I noticed that there are 1977 terms to add, and of course it would not be math contest if there was not some interesting approach to find this sum without putting 1977 over the common denominator. þ pffiffiffiffiffiffiffi 1977þ 1978 I knew that this sum exists because they would not ask to evaluate it otherwise.
I remember that one of the problems looked like this. Based on my own experience, the first time I had to consider a series, different from an arithmetic or geometric progression, it was at my city math olympiad when I was in 9th grade.
Each year contest problems become more and more challenging and even an “A” math student without special preparation for such problems could feel frustrated and lost.
SOLVING SEQUENCES SERIES
Printed on acid-free paper This book is published under the trade name Birkha¨user, The registered company is Springer International Publishing AG The registered company address is: GewerbestraCham, Switzerland To my Beautiful Daughter, Sasha And to my Wonderful Parents, Natali and Valery Grigoriev Your Encouragement Made this Book Possible And to my University Mentor and Scientific Advisor Academician, Stepanov Nikolay Fedorovich Without your Help and Brilliant Mind my Career as a Scientist Would not be Successful! Preface Many contest problems concern series and sequences. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication.
SOLVING SEQUENCES FREE
in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The use of general descriptive names, registered names, trademarks, service marks, etc. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.
Table of contents : Front Matter.Pages i-xx Introduction to Sequences and Series.Pages 1-63 Further Study of Sequences and Series.Pages 65-121 Series Convergence Theorems and Applications.Pages 123-190 Real-Life Applications of Geometric and Arithmetic Sequences.Pages 191-225 Homework.Pages 227-269 Back Matter.Pages 271-281 Citation previewĮllina Grigorieva Methods of Solving Sequence and Series Problems Ellina Grigorieva Methods of Solving Sequence and Series Problems Ellina Grigorieva Department of Mathematics and Computer Science Texas Woman’s University Denton, TX, USA ISBN 978-5-0 ISBN 978-6-7 DOI 10.1007/978-6-7 (eBook) Library of Congress Control Number: 2016952549 Mathematics Subject Classification (2010): 97I30, 97U40, 97I50, 97I40 © Springer International Publishing Switzerland 2016 This work is subject to copyright.
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