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Difference between revisions of "Leibniz series"

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The alternating series
 
The 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/l058110/l0581101.png" /></td> </tr></table>
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1-\frac13+\frac15-\frac17+\dots,
 
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\end{equation}
which converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058110/l0581102.png" />. It was considered by G. Leibniz in 1673–1674.
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which converges to $\pi/4$. It was considered by G. Leibniz in 1673–1674.
 
 
 
 
 
 
====Comments====
 
 
 
  
 
====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>
 
<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>

Revision as of 09:36, 12 December 2012

The alternating series \begin{equation} 1-\frac13+\frac15-\frac17+\dots, \end{equation} which converges to $\pi/4$. It was considered by G. Leibniz in 1673–1674.

References

[a1] K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990)
How to Cite This Entry:
Leibniz series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Leibniz_series&oldid=29173
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article