# Lambert summation method

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=30239