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)

    Parents ought, through their own behavior and the values by which they live, to provide direction for their children. But they need to rid themselves of the idea that there are surefire methods which, when well applied, will produce certain predictable results. Whatever we do with and for our children ought to flow from our understanding of and our feelings for the particular situation and the relation we wish to exist between us and our child.
    Bruno Bettelheim (20th century)