Dilution of a 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

\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).