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Difference between revisions of "Leibniz criterion"

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''for convergence of an alternating series''
 
''for convergence of an alternating series''
  
 
If the terms of an alternating series
 
If the terms of an alternating series
 
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\begin{equation}
<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/l/l058/l058090/l0580901.png" /></td> </tr></table>
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\sum_{n=1}^{\infty}(-1)^{n+1}a_n,\quad a_n>0,
 
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\end{equation}
decrease monotonically (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058090/l0580902.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058090/l0580903.png" />) and tend to zero (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058090/l0580904.png" />), then the series converges; moreover, a remainder of the series,
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decrease monotonically ($a_n\geq a_{n+1}$, $n=1,2,\dots$) and tend to zero ( $\lim\limits_{n\to\infty}a_n=0$ ), then the series converges; moreover, a remainder of the series,
 
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\begin{equation}
<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/l/l058/l058090/l0580905.png" /></td> </tr></table>
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\sum_{k=n+1}^{\infty}(-1)^{k+1}a_k,
 
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\end{equation}
 
has the sign of its first term and is less than it in absolute value. The criterion was established by G. Leibniz in 1682.
 
has the sign of its first term and is less than it in absolute value. The criterion was established by G. Leibniz in 1682.
  
  
 
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====Examples====
====Comments====
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* [[Leibniz_series|Leibniz series]] $\sum\limits_{n=1}^{\infty}\frac{(-1)^{n+1}}{2n-1}$.
 
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Knopp,  "Theorie und Anwendung der unendlichen Reihen" , Springer  (1964)  (English translation: Blackie, 1951 &amp; Dover, reprint, 1990)</TD></TR></table>
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|valign="top"|{{Ref|Kn}}|| K. Knopp,  "Theorie und Anwendung der unendlichen Reihen" , Springer  (1964)  (English translation: Blackie, 1951 &amp; Dover, reprint, 1990)
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Latest revision as of 20:16, 9 December 2013

2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]

for convergence of an alternating series

If the terms of an alternating series \begin{equation} \sum_{n=1}^{\infty}(-1)^{n+1}a_n,\quad a_n>0, \end{equation} decrease monotonically ($a_n\geq a_{n+1}$, $n=1,2,\dots$) and tend to zero ( $\lim\limits_{n\to\infty}a_n=0$ ), then the series converges; moreover, a remainder of the series, \begin{equation} \sum_{k=n+1}^{\infty}(-1)^{k+1}a_k, \end{equation} has the sign of its first term and is less than it in absolute value. The criterion was established by G. Leibniz in 1682.


Examples

References

[Kn] K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990)
How to Cite This Entry:
Leibniz criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Leibniz_criterion&oldid=18694
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article