Axiomatic Methods
Taking regularity, linearity and stability as axioms, it is possible to sum many divergent series by elementary algebraic manipulations. For instance, whenever r ≠ 1, 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 ∞.
Read more about this topic: Divergent Series
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