Difference between revisions of "Logarithmic summation method"
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One of the methods for summing series of numbers. A series with partial sums s_n is summable by the logarithmic method to the sum s if the logarithmic mean | 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 | ||
| − | $$\sigma_m=\frac{1}{\sum_{k=0}^m\frac{1}{k+1}}\left(s_0+\frac{s_1}{2}+\ | + | $$\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|Riesz summation method]] (R,p_n) with p_n=1/(n+1). It is equivalent to and compatible (cf. [[Compatibility of summation methods|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|Arithmetical averages, summation method of]]). | converges to s as m\to\infty. The logarithmic summation method is the [[Riesz summation method|Riesz summation method]] (R,p_n) with p_n=1/(n+1). It is equivalent to and compatible (cf. [[Compatibility of summation methods|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|Arithmetical averages, summation method of]]). | ||
Latest revision as of 13:14, 14 February 2020
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) |
How to Cite This Entry:
Logarithmic summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logarithmic_summation_method&oldid=32663
Logarithmic summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logarithmic_summation_method&oldid=32663
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article