terms of the sequence, the average converges, and we can use this average instead of a limit to evaluate the sum of the series. A summation method can be seen as a function from a set of sequences of partial sums to values. If A is any summation method assigning values to a set of sequences, we may mechanically translate this to a series-summation method AΣ that assigns the same values to the corresponding series. There are certain properties it is desirable for these methods to possess if they are to arrive at values corresponding to limits and sums, respectively. Regularity. A summation method is regular if, whenever the sequence s converges to x, Equivalently, the corresponding series-summation method evaluates Linearity. A is linear if it is a linear functional on the sequences where it is defined, so that for sequences r, s and a real or complex scalar k. Since the terms of the series a are linear functionals on the sequence s and vice versa, this is equivalent to AΣ being a linear functional on the terms of the series. Stability (also called translativity). If s is a sequence starting from s0 and s′ is the sequence obtained by omitting the first value and subtracting it from the rest, so that , then A(s) is defined if and only if A(s′) is defined, and Equivalently, whenever for all n, then Another way of stating this is that the shift rule must be valid for the series that are summable by this method. The third condition is less important, and some significant methods, such as Borel summation, do not possess it. One can also give a weaker alternative to the last condition. Finite re-indexability. If a and a′ are two series such that there exists a bijection such that for all i, and if there exists some such that for all i > N, then (In other words, a′ is the same series as a, with only finitely many terms re-indexed.) This is a weaker condition than stability, because any summation method that exhibits stability also exhibits finite re-indexability, but the converse is not true.) A desirable property for two distinct summation methods A and B to share is consistency: A and B are consistent if for every sequence s to which both assign a value, (Using this language, a summation method A is regular iff it is consistent