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One of the methods for the summation of series of numbers. The series
 
One of the methods for the summation of series of numbers. The series
 
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$$
<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/a010/a010170/a0101701.png" /></td> </tr></table>
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\sum_{k=0}^\infty a_k
 
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$$
can be summed by the Abel method (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010170/a0101703.png" />-method) to the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010170/a0101704.png" /> if, for any real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010170/a0101705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010170/a0101706.png" />, the series
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can be summed by the Abel method ($A$-method) to the number $A$ if, for any real $x$, $0<x<1$, the series
 
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$$
<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/a010/a010170/a0101707.png" /></td> </tr></table>
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\sum_{k=0}^\infty a_k x^k
 
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$$
 
is convergent and
 
is convergent and
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$$
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\lim_{x\rightarrow 1-0} \sum_{k=0}^\infty a_k x^k = S.
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$$
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This summation method can already be found in the works of L. Euler and G. Leibniz. The name "Abel summation method" originates from the [[Abel theorem|Abel theorem]] on the continuity of power series. The Abel summation method belongs to the class of totally [[Regular summation methods|regular summation methods]] and is more powerful than the entire set of [[Cesàro summation methods|Cesàro summation methods]]. The Abel summation method is used in conjunction with Tauberian theorems to demonstrate the convergence of a 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/a010/a010170/a0101708.png" /></td> </tr></table>
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A closely related summation method is the $A^*$-method. Let $z$ be a complex number, $\left|z\right|<1$; the series
 
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$$
This summation method can already be found in the works of L. Euler and G. Leibniz. The name  "Abel summation method"  originates from the [[Abel theorem|Abel theorem]] on the continuity of power series. The Abel summation method belongs to the class of totally [[Regular summation methods|regular summation methods]] and is more powerful than the entire set of [[Cesàro summation methods|Cesàro summation methods]]. The Abel summation method is used in conjunction with Tauberian theorems to demonstrate the convergence of a series.
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\sum_{k=0}^\infty a_k
 
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$$
A closely related summation method is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010170/a0101709.png" />-method. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010170/a01017010.png" /> be a complex number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010170/a01017011.png" />; the series
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is summed by the $A^*$-method to the number $S$ if
 
+
$$
<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/a010/a010170/a01017012.png" /></td> </tr></table>
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\lim \sum_{k=0}^\infty a_k z^k = S,
 
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$$
is summed by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010170/a01017014.png" />-method to the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010170/a01017015.png" /> if
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where $z\rightarrow 1$ along any path not tangent to the unit circle.
 
 
<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/a010/a010170/a01017016.png" /></td> </tr></table>
 
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010170/a01017017.png" /> along any path not tangent to the unit circle.
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====References====
  
====References====
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{|
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,  "Divergent series" , Clarendon Press  (1949)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,   "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)</TD></TR></table>
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|-
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|valign="top"|{{Ref|Ba}}||valign="top"| N.K. [N.K. Bari] Bary, "A treatise on trigonometric series", Pergamon (1964) (Translated from Russian)
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|-
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|valign="top"|{{Ref|Ha}}||valign="top"| G.H. Hardy, "Divergent series", Clarendon Press (1949)
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|-
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|}

Revision as of 17:33, 7 May 2012


One of the methods for the summation of series of numbers. The series $$ \sum_{k=0}^\infty a_k $$ can be summed by the Abel method ($A$-method) to the number $A$ if, for any real $x$, $0<x<1$, the series $$ \sum_{k=0}^\infty a_k x^k $$ is convergent and $$ \lim_{x\rightarrow 1-0} \sum_{k=0}^\infty a_k x^k = S. $$ This summation method can already be found in the works of L. Euler and G. Leibniz. The name "Abel summation method" originates from the Abel theorem on the continuity of power series. The Abel summation method belongs to the class of totally regular summation methods and is more powerful than the entire set of Cesàro summation methods. The Abel summation method is used in conjunction with Tauberian theorems to demonstrate the convergence of a series.

A closely related summation method is the $A^*$-method. Let $z$ be a complex number, $\left|z\right|<1$; the series $$ \sum_{k=0}^\infty a_k $$ is summed by the $A^*$-method to the number $S$ if $$ \lim \sum_{k=0}^\infty a_k z^k = S, $$ where $z\rightarrow 1$ along any path not tangent to the unit circle.

References

[Ba] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series", Pergamon (1964) (Translated from Russian)
[Ha] G.H. Hardy, "Divergent series", Clarendon Press (1949)
How to Cite This Entry:
Abel summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abel_summation_method&oldid=26200
This article was adapted from an original article by A.A. Zakharov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article