Namespaces
Variants
Actions

Difference between revisions of "Dilution of a series"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoding is done)
(cf Summation methods)
 
Line 10: Line 10:
 
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 (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 $a\ne s$).
+
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 22:41, 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://www.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