# Translativity of a summation method

The property of the method consisting in the preservation of summability of a series after adding to or deleting from it a finite number of terms. More precisely, a summation method $\mathcal{A}$ is said to be translative if the summability of the series $$\sum_{k=0}^\infty a_k$$ to the sum $S_1$ implies that the series $$\sum_{k=1}^\infty a_k$$ is summable by the same method to the sum $S_1 - a_0$, and conversely. For a summation method $\mathcal{A}$ defined by transformation of the sequence $S_n$ into a sequence or function, the property of translativity consists of the equivalence of the conditions $$\mathcal{A}\text{-}\lim S_n = S$$ and $$\mathcal{A}\text{-}\lim S_{n+1} = S$$
If the summation method is defined by a regular matrix $(A_{nk})$ (cf. Regular summation methods), then this means that $$\label{eq:a1} \lim_{n\rightarrow\infty} \sum_{k=0}^\infty A_{nk} S_k = S$$ always implies that $$\label{eq:a2} \lim_{n\rightarrow\infty} \sum_{k=0}^\infty A_{nk} S_{k+1} = S$$ and conversely. In cases when such an inference only holds in one direction, the method is called right translative if \ref{eq:a1} implies \ref{eq:a2} but the converse is false, or left translative if \ref{eq:a2} implies \ref{eq:a1} but the converse is false.
Many widely used summation methods have the property of translativity; for example, the Cesàro summation methods $(C,k)$ for $k > 0$, the Riesz summation method $R(n,k)$ for $k>0$ and the Abel summation method are translative; the Borel summation method is left translative.