Linear and stable summation methods

Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $M$ be a summability method. Write $M\left(\sum a_n\right)$ for the $M$-sum of an $M$-summable series $\sum_{n \ge 0} a_n$ with terms in $\mathbb{K}$.

1. $M$ is linear if, whenever $\sum a_n$ and $\sum b_n$ are $M$-summable and $\lambda, \mu \in \mathbb{K}$, the series $\sum (\lambda a_n + \mu b_n)$ is $M$-summable and $M\left(\sum (\lambda a_n + \mu b_n)\right) = \lambda M\left(\sum a_n\right) + \mu M\left(\sum b_n\right)$.
2. $M$ is stable if, for every sequence $(a_n)_{n \ge 0}$, the series $a_0 + a_1 + a_2 + \cdots$ is $M$-summable if and only if $a_1 + a_2 + \cdots$ is $M$-summable, and then $M(a_0 + a_1 + a_2 + \cdots) = a_0 + M(a_1 + a_2 + \cdots)$.