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Alternating series

From Encyclopedia of Mathematics
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An infinite series whose terms are alternately positive and negative:

If the terms of an alternating series are monotone decreasing and tend to zero , then the series is convergent (Leibniz' theorem). The remainder term of an alternating series,

has the same sign as its first term and is less then the latter in absolute value. The simplest examples of alternating series are

The sum of the first of these series is ; that of the second is .

How to Cite This Entry:
Alternating series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Alternating_series&oldid=25988
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article