Divergent Series - Axiomatic Methods

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

\begin{align} G(r,c) & = \sum_{k=0}^\infty cr^k & & \\ & = c + \sum_{k=0}^\infty cr^{k+1} & & \mbox{ (stability) } \\ & = c + r \sum_{k=0}^\infty cr^k & & \mbox{ (linearity) } \\ & = c + r \, G(r,c), & & \mbox{ whence } \\ G(r,c) & = \frac{c}{1-r}, & & \\ \end{align}

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 ∞.

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