Logarithmic summation method

One of the methods for summing series of numbers. A series $\sum_{k=0}^\infty a_k$ with partial sums $s_n$ is summable by the logarithmic method to the sum $s$ if the logarithmic mean

$$\sigma_m=\frac{1}{\sum_{k=0}^m\frac{1}{k+1}}\left(s_0+\frac{s_1}{2}+\dotsb+\frac{s_m}{m+1}\right)$$

converges to $s$ as $m\to\infty$. The logarithmic summation method is the Riesz summation method $(R,p_n)$ with $p_n=1/(n+1)$. It is equivalent to and compatible (cf. Compatibility of summation methods) with the Riesz summation method $(R,\lambda_n,1)$ with $\lambda_n=\ln(n+1)$ and is more powerful than the summation method of arithmetical averages (cf. Arithmetical averages, summation method of).

References

[1]	F. Riesz, "Sur la sommation des séries de Dirichlet" C.R. Acad. Sci. Paris , 149 (1909) pp. 18–21
[2]	G.H. Hardy, "Divergent series" , Clarendon Press (1949)