# 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 (after dilution a 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 $a\ne s$). |

## Revision as of 11:44, 17 October 2013

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 $a\ne s$).

**How to Cite This Entry:**

Dilution of a series.

*Encyclopedia of Mathematics.*URL: http://www.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