Linear and stable summation methods

Summary

A summation method is linear when it respects sums and scalar multiples of series. It is stable when deleting or adding a first term changes the sum by that term. These two properties, together with regularity, are the ones usually expected of a reasonable method. They are strong enough to force values: any linear and stable method that sums $1 - 1 + 1 - \cdots$ must give it the value $\frac12$.