Difference between revisions of "Lambert summation method"
From Encyclopedia of Mathematics
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| − | A method for summing series of numbers. The series | + | 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 |
| − | + | $$ | |
| − | + | \sum_{n=1}^\infty a_n | |
| − | + | $$ | |
| − | is summable by Lambert's method to the number | + | is summable by Lambert's method to the number $A$ if |
| − | + | $$ | |
| − | + | \lim_{y \downto 0} F(y) = A | |
| − | + | $$ | |
where | 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 [[#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. | ||
| − | |||
| − | |||
====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> | + | <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> | ||
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
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