Matrix summability method

Remarks

Example: the Cesàro matrix has $a_{nk} = \frac{1}{n+1}$ for $k \le n$ and $a_{nk} = 0$ for $k > n$. For $s = (1, 0, 1, 0, \dots)$ one has $(As)_n = \frac{\lceil (n+1)/2 \rceil}{n+1} \to \frac12$, so $s$ is $A$-summable to $\frac12$. Non-example: for $a_{nk} = 2^{-k-1}$ and $s_k = 2^k$, the series $\sum_k a_{nk} s_k = \sum_k \frac12$ diverges, so $(As)_n$ is not defined and $s$ is not $A$-summable. The identity matrix gives ordinary convergence. Every matrix method is linear, because $A(\lambda s + \mu t) = \lambda As + \mu At$.