Namespaces
Variants
Actions

Difference between revisions of "Dilution of a series"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoding is done)
Line 1: Line 1:
 
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032440/d0324401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
\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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032440/d0324402.png" /></td> </tr></table>
+
\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 (*) summable to the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032440/d0324403.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032440/d0324404.png" />).
+
\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 10:44, 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