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 ∞.

Read more about this topic:  Divergent Series

Famous quotes containing the words axiomatic and/or methods:

    “It is ... axiomatic that we should all think of ourselves as being more sensitive than other people because, when we are insensitive in our dealings with others, we cannot be aware of it at the time: conscious insensitivity is a self-contradiction.”
    —W.H. (Wystan Hugh)

    “A writer who writes, “I am alone” ... can be considered rather comical. It is comical for a man to recognize his solitude by addressing a reader and by using methods that prevent the individual from being alone. The word alone is just as general as the word bread. To pronounce it is to summon to oneself the presence of everything the word excludes.”
    —Maurice Blanchot (b. 1907)