# 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 $$\tag{1} \lim_{n\rightarrow\infty} \sum_{k=0}^\infty A_{nk} S_k = S$$ always implies that $$\tag{2} \lim_{n\rightarrow\infty} \sum_{k=0}^\infty A_{nk} S_{k+1} = S$$ and conversely. In cases when such an inference only holds in one direction, the method is called right translative if (1) implies (2) but the converse is false, or left translative if (2) implies (1) 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.