with the standard sum Σ.) If two methods are consistent, and one sums more series than the other, the one summing more series is stronger. There are powerful numerical summation methods that are neither regular nor linear, for instance nonlinear sequence transformations like Levin-type sequence transformations and Padé approximants, as well as the order-dependent mappings of perturbative series based on renormalization techniques. Taking regularity, linearity and stability as axioms, it is possible to sum many divergent series by elementary algebraic manipulations. This partly explains why many different summation methods give the same answer for certain series. For instance, whenever the geometric series can be evaluated regardless of convergence. More rigorously, any summation method that possesses these properties and which assigns a finite value to the geometric series must assign this value. However, when r is a real number larger than 1, the partial sums increase without bound, and averaging methods assign a limit of infinity. Classical summation methods The two classical summation methods for series, ordinary convergence and absolute convergence, define the sum as a limit of certain partial sums. These are included only for completeness; strictly speaking they are not true summation methods for divergent series since, by definition, a series is divergent only if these methods do not work. Most but not all summation methods for divergent series extend these methods to a larger class of sequences. Absolute convergence Absolute convergence defines the sum of a sequence (or set) of numbers to be the limit of the net of all partial sums , if it exists. It does not depend on the order of the elements of the sequence, and a classical theorem says that a sequence is absolutely convergent if and only if the sequence of absolute values is convergent in the standard sense. Sum of a series Cauchy's classical definition of the sum of a series defines the sum to be the limit of the sequence of partial sums . This is the default definition of convergence of a sequence. Nørlund means Suppose pn is a sequence of positive terms, starting from p0. Suppose also that If now we transform a sequence s by using p to give weighted means, setting then the limit of tn as n goes to infinity is an average called the Nørlund mean Np(s). The Nørlund mean is regular, linear, and stable. Moreover, any two Nørlund means are consistent. Cesàro summation