# 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.