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Difference between revisions of "Lambert summation method"

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A method for summing series of numbers. The series
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A [[Regular summation methods|summation method]] for summing series of complex numbers which assigns a sum to certain [[divergent series]] as well as those which are [[convergent series|convrgent]] in the usual sense]]. 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/l/l057/l057350/l0573501.png" /></td> </tr></table>
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\sum_{n=1}^\infty a_n
 
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$$
is summable by Lambert's method to the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057350/l0573502.png" /> if
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is summable by Lambert's method to the number $A$ if
 
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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/l/l057/l057350/l0573503.png" /></td> </tr></table>
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\lim_{y \downto 0} F(y) = A
 
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$$
 
where
 
where
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$$
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F(y) = \sum_{n=1}^\infty a_n \frac{n y \exp(-ny)}{1-exp(-ny)}
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$$
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for $y>0$, if the series on the right-hand side converges. The method was proposed by J.H. Lambert [[#References|[1]]]. The summability of a series by the Cesàro summation method $(C,k) for some $k > -1$ (cf. [[Cesàro summation methods|Cesàro summation methods]]) to the sum $A$ implies its summability by the Lambert method to the same sum, and if the series is summable by the Lambert method to the sum $A$, then it is also summable by the [[Abel summation method|Abel summation method]] to the same sum.
  
<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/l057/l057350/l0573504.png" /></td> </tr></table>
 
  
and if the series on the right-hand side converges. The method was proposed by J.H. Lambert [[#References|[1]]]. The summability of a series by the Cesàro summation method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057350/l0573505.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057350/l0573506.png" /> (cf. [[Cesàro summation methods|Cesàro summation methods]]) to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057350/l0573507.png" /> implies its summability by the Lambert method to the same sum, and if the series is summable by the Lambert method to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057350/l0573508.png" />, then it is also summable by the [[Abel summation method|Abel summation method]] to the same sum. Lambert's summation method is regular (see [[Regular summation methods|Regular summation methods]]).
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.H. Lambert,  "Anlage zur Architektonik" , '''2''' , Riga  (1771)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.H. Hardy,  "Divergent series" , Clarendon Press  (1949)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  J.H. Lambert,  "Anlage zur Architektonik" , '''2''' , Riga  (1771)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  G.H. Hardy,  "Divergent series" , Clarendon Press  (1949)</TD></TR>
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</table>

Revision as of 19:00, 25 August 2013

A summation method for summing series of complex numbers which assigns a sum to certain divergent series as well as those which are convrgent in the usual sense]]. The series $$ \sum_{n=1}^\infty a_n $$ is summable by Lambert's method to the number $A$ if $$ \lim_{y \downto 0} F(y) = A $$ where $$ F(y) = \sum_{n=1}^\infty a_n \frac{n y \exp(-ny)}{1-exp(-ny)} $$ for $y>0$, if the series on the right-hand side converges. The method was proposed by J.H. Lambert [1]. The summability of a series by the Cesàro summation method $(C,k) for some $k > -1$ (cf. [[Cesàro summation methods|Cesàro summation methods]]) to the sum $A$ implies its summability by the Lambert method to the same sum, and if the series is summable by the Lambert method to the sum $A$, then it is also summable by the Abel summation method to the same sum.


References

[1] J.H. Lambert, "Anlage zur Architektonik" , 2 , Riga (1771)
[2] G.H. Hardy, "Divergent series" , Clarendon Press (1949)
How to Cite This Entry:
Lambert summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lambert_summation_method&oldid=14975
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article