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Difference between revisions of "Alternating series"

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An infinite series whose terms are alternately positive and negative:
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{{MSC|40A05}}
 
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If the terms of an alternating series are monotone decreasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012050/a0120502.png" /> and tend to zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012050/a0120503.png" />, then the series is convergent (Leibniz' theorem). The remainder term of an alternating series,
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012050/a0120504.png" /></td> </tr></table>
 
  
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An alternating series is an infinite series whose terms are alternately positive and negative:
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$$
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u_1 - u_2 + \cdots + (-1)^{n-1}u_n + \cdots, \quad u_k > 0.
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$$
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If the terms of an alternating series are monotone decreasing $(u_{n+1} < u_n)$ and tend to zero ($\lim_{n\rightarrow\infty} u_n = 0$), then the series is convergent (Leibniz' theorem). The remainder term of an alternating series,
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$$
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r_n = (-1)^{n-1}u_n +\cdots,
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$$
 
has the same sign as its first term and is less then the latter in absolute value. The simplest examples of alternating series are
 
has the same sign as its first term and is less then the latter in absolute value. The simplest examples of alternating series are
 
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$$
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1 -
 
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\frac{1}{2} +
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\frac{1}{3} -
 
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\frac{1}{4} + \cdots +
The sum of the first of these series is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012050/a0120507.png" />; that of the second is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012050/a0120508.png" />.
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(-1)^{n-1} \frac{1}{n} + \cdots
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$$
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and
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$$
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1 -
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\frac{1}{3} +
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\frac{1}{5} -
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\frac{1}{7} + \cdots +
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(-1)^{n-1} \frac{1}{2n-1} + \cdots.
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$$
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The sum of the first of these series is $\log 2$; that of the second is $\pi/4$.

Latest revision as of 18:34, 4 May 2012

2010 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]

An alternating series is an infinite series whose terms are alternately positive and negative: $$ u_1 - u_2 + \cdots + (-1)^{n-1}u_n + \cdots, \quad u_k > 0. $$ If the terms of an alternating series are monotone decreasing $(u_{n+1} < u_n)$ and tend to zero ($\lim_{n\rightarrow\infty} u_n = 0$), then the series is convergent (Leibniz' theorem). The remainder term of an alternating series, $$ r_n = (-1)^{n-1}u_n +\cdots, $$ has the same sign as its first term and is less then the latter in absolute value. The simplest examples of alternating series are $$ 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots + (-1)^{n-1} \frac{1}{n} + \cdots $$ and $$ 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots + (-1)^{n-1} \frac{1}{2n-1} + \cdots. $$ The sum of the first of these series is $\log 2$; that of the second is $\pi/4$.

How to Cite This Entry:
Alternating series. Encyclopedia of Mathematics. URL: http://www.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