Difference between revisions of "Dilution of a series"
From Encyclopedia of Mathematics
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The inclusion of any finite number of zeros between adjacent terms of a series. For the series | The inclusion of any finite number of zeros between adjacent terms of a series. For the series | ||
| − | + | \begin{equation}\label{eq:1} | |
| + | \sum\limits_{k=0}^{\infty}u_k | ||
| + | \end{equation} | ||
a diluted series has the form | a diluted series has the form | ||
| − | + | \begin{equation} | |
| − | + | u_0+0+\dots+0+u_1+0+\dots+0+u_2+\dots | |
| − | Dilution of a series does not affect convergence of the series, but it may violate summability of the series | + | \end{equation} |
| + | Dilution of a series does not affect convergence of the series, but it may violate summability of the series: after dilution a series \eqref{eq:1} summable to the number $s$ by some summation method may turn out to be not summable at all by this method or may turn out to be summable to a number $a\ne s$ (cf [[Summation methods]]). | ||
Latest revision as of 20:41, 14 October 2014
The inclusion of any finite number of zeros between adjacent terms of a series. For the series
\begin{equation}\label{eq:1} \sum\limits_{k=0}^{\infty}u_k \end{equation}
a diluted series has the form
\begin{equation} u_0+0+\dots+0+u_1+0+\dots+0+u_2+\dots \end{equation} Dilution of a series does not affect convergence of the series, but it may violate summability of the series: after dilution a series \eqref{eq:1} summable to the number $s$ by some summation method may turn out to be not summable at all by this method or may turn out to be summable to a number $a\ne s$ (cf Summation methods).
How to Cite This Entry:
Dilution of a series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dilution_of_a_series&oldid=18098
Dilution of a series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dilution_of_a_series&oldid=18098
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article