Consistency of summability methods

Remarks

Example: Cesàro summation and Abel summation are consistent. Indeed, if $\sum a_n$ is Cesàro summable to $s$, then it is Abel summable to the same $s$, so whenever both methods apply their values agree.

Non-example: let $A$ assign to a series the limit of its even partial sums $s_0, s_2, s_4, \dots$, and let $B$ assign the limit of its odd partial sums $s_1, s_3, s_5, \dots$. Both methods give the usual sum to every convergent series. For $1 - 1 + 1 - 1 + \cdots$, the even partial sums are all $1$ and the odd ones are all $0$. So both methods sum this series, with values $1$ and $0$, and $A$ and $B$ are not consistent.

The same notion applies to methods that act on sequences rather than series, by comparing the values they assign to a common summable sequence.