# Alternating Series

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## alternating series

[′ȯl·tər·nād·iŋ ′sir·ēz] (mathematics)

Any series of real numbers in which consecutive terms have opposite signs.

McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

The following article is from

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.## Alternating Series

an infinite series whose terms are alternately positive and negative:

*u _{1}* -

*u*+

_{2}*u*-

_{3}*u*+ … + (-I)

_{4}^{n-1}

*u*

_{n}+ …

*for u _{k}* > 0. If the terms of an alternating series monotonically decrease (

*u*<

_{n+1}*u*) and tend toward zero (lim

_{n}*u*= 0), then the series is convergent (the Leibniz theorem). The remainder of the convergent series

_{n}*r _{n}* = (-1)

^{n}u

_{n+1}+ …

has the sign of its first term and is less than this term in absolute value. Some very simple examples of convergent alternating series are

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.