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Table of Integrals, Series, and Products
Seventh Edition

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Table of Integrals, Series, and Products
Seventh Edition

I.S. Gradshteyn and I.M. Ryzhik
Alan Jeffrey, Editor
University of Newcastle upon Tyne, England

Daniel Zwillinger, Editor
Rensselaer Polytechnic Institute, USA

Translated from Russian by Scripta Technica, Inc.

AMSTERDAM • BOSTON • HEIDELBERG • LONDON
NEW YORK • OXFORD • PARIS • SAN DIEGO
SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
Academic Press is an imprint of Elsevier

Academic Press is an imprint of Elsevier
30 Corporate Drive, Suite 400, Burlington, MA 01803, USA
525 B Street, Suite 1900, San Diego, California 92101-4495, USA
84 Theobald’s Road, London WC1X 8RR, UK
This book is printed on acid-free paper.

∞

Copyright c 2007, Elsevier Inc. All rights reserved.
No part of this publication may be reproduced or transmitted in any form or by any
means, electronic or mechanical, including photocopy, recording, or any information
storage and retrieval system, without permission in writing from the publisher.
Permissions may be sought directly from Elsevier’s Science & Technology Rights
Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333,
E-mail: permissions@elsevier.com. You may also complete your request online
via the Elsevier homepage (), by selecting “Support & Contact”
then “Copyright and Permission” and then “Obtaining Permissions.”
For information on all Elsevier Academic Press publications
visit our Web site at 
ISBN-13: 978-0-12-373637-6
ISBN-10: 0-12-373637-4
PRINTED IN THE UNITED STATES OF AMERICA
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Contents
Preface to the Seventh Edition
Acknowledgments
The Order of Presentation of the Formulas
Use of the Tables
Index of Special Functions
Notation
Note on the Bibliographic References
0

Introduction
0.1
0.11
0.12
0.13
0.14
0.15
0.2
0.21
0.22
0.23–0.24
0.25
0.26
0.3
0.30
0.31
0.32
0.33
0.4
0.41
0.42
0.43
0.44

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xxiii
xxvii
xxxi
xxxix
xliii
xlvii

Finite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sums of powers of natural numbers . . . . . . . . . . . . . . . .
Sums of reciprocals of natural numbers . . . . . . . . . . . . . .
Sums of products of reciprocals of natural numbers . . . . . . .
Sums of the binomial coefficients . . . . . . . . . . . . . . . . .
Numerical Series and Infinite Products . ...
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