# Difference between revisions of "Dilution of a series"

From Encyclopedia of Mathematics

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u_0+0+\dots+0+u_1+0+\dots+0+u_2+\dots | u_0+0+\dots+0+u_1+0+\dots+0+u_2+\dots | ||

\end{equation} | \end{equation} | ||

− | Dilution of a series does not affect convergence of the series, but it may violate summability of the series | + | 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=30623

This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article