Summability method

Remarks

Example: the ordinary limit, with $D_M$ the set of convergent sequences and $M(s) = \lim s_n$, is a summability method; it sums exactly the convergent series. Example: Cesàro summation takes $D_M$ to be the sequences whose Cesàro means converge, and $M(s)$ the limit of these means; it sums $1 - 1 + 1 - \cdots$ to $\frac12$. Non-example: "take the limit of some convergent subsequence" is not a summability method. The sequence $0, 1, 0, 1, \dots$ would get both $0$ and $1$, so the rule does not define a map. Some authors define methods directly on series $(a_n)$ instead of on partial sums; the two points of view are equivalent.