Difference between revisions of "Logarithmic summation method"
From Encyclopedia of Mathematics
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| − | One of the methods for summing series of numbers. A series | + | {{TEX|done}} |
| + | 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 | ||
| − | + | $$\sigma_m=\frac{1}{\sum_{k=0}^m\frac{1}{k+1}}\left(s_0+\frac{s_1}{2}+\ldots+\frac{s_m}{m+1}\right)$$ | |
| − | converges to | + | 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]]). |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> F. Riesz, "Sur la sommation des séries de Dirichlet" ''C.R. Acad. Sci. Paris'' , '''149''' (1909) pp. 18–21</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"> F. Riesz, "Sur la sommation des séries de Dirichlet" ''C.R. Acad. Sci. Paris'' , '''149''' (1909) pp. 18–21</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 15:07, 1 August 2014
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
$$\sigma_m=\frac{1}{\sum_{k=0}^m\frac{1}{k+1}}\left(s_0+\frac{s_1}{2}+\ldots+\frac{s_m}{m+1}\right)$$
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