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In [[mathematical analysis]], '''Lambert summation''' is a summability method for a class of [[divergent series]].  
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==Definition==
A series <math>\sum a_n</math> is ''Lambert summable'' to ''A'', written <math>\sum a_n = A \,(\mathrm{L})</math>, if
 
:<math>\lim_{r \rightarrow 1-} (1-r) \sum_{n=1}^\infty \frac{n a_n r^n}{1-r^n} = A . \, </math>
 
If a series is convergent to ''A'' then it is Lambert summable to ''A'' (an [[Abelian theorem]]).
 
==Examples==
 
* <math>\sum_{n=1}^\infty \frac{\mu(n)}{n} = 0 (\mathrm{L})</math>, where &mu; is the [[Möbius function]]. Hence if this series converges at all, it converges to zero.
 
==See also==
* [[Lambert series]]
* [[Abel–Plana formula]]
* [[Abelian and tauberian theorems]]
 
==References==
* {{cite book | author=Jacob Korevaar | title=Tauberian theory. A century of developments | series=Grundlehren der Mathematischen Wissenschaften | volume=329 | publisher=[[Springer-Verlag]] | year=2004 | isbn=3-540-21058-X | pages=18 }}
*{{cite book | author=Hugh L. Montgomery | authorlink=Hugh Montgomery (mathematician) | coauthors=[[Robert Charles Vaughan (mathematician)|Robert C. Vaughan]] | title=Multiplicative number theory I. Classical theory | series=Cambridge tracts in advanced mathematics | volume=97 | year=2007 | isbn=0-521-84903-9 | pages=159–160 | publisher=Cambridge Univ. Press | location=Cambridge}}
*{{cite journal | author=Norbert Wiener | authorlink=Norbert Wiener | title=Tauberian theorems | journal=Ann. Of Math. | year=1932 | volume=33 | pages=1–100 | doi=10.2307/1968102 | issue=1 | publisher=The Annals of Mathematics, Vol. 33, No. 1 | jstor=1968102 }}
 
{{Mathanalysis-stub}}
 
[[Category:Mathematical series]]
[[Category:Summability methods]]

Latest revision as of 11:34, 9 August 2014

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