Methods of Solving Sequence and Series Problems
This book aims to dispel the mystery and fear experienced by students surrounding sequences, series, convergence, and their applications. The author, an accomplished female mathematician, achieves this by taking a problem solving approach, starting with f
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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 978-3-319-45685-0 ISBN 978-3-319-45686-7 DOI 10.1007/978-3-319-45686-7
(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. 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. The use of general descriptive names, registered names, trademarks, service marks, etc. 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 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. 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. Printed on acid-free paper This book is published under the trade name Birkha¨user, www.birkhauser-science.com The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, 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. 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. 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. I remember that one of the problems looked like this. 1 pffiffiffiffiffiffiffi Evaluate the sum 1þ1pffiffi2 þ pffiffi2þ1 pffiffi3 þ pffiffi3þ1 pffiffi4 þ . . . þ pffiffiffiffiffiffiffi . 1977þ 1978
I knew that this sum exists because they would not ask to evaluate it otherwise. 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. The first thing I tried worked. I multiplied the numerator and denominato
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