Background
One may study the convergence of series whose terms an are elements of an arbitrary abelian topological group. The notion of absolute convergence requires more structure, namely a norm, which is a real-valued function on abelian group G (written additively, with identity element 0) such that:
- The norm of the identity element of G is zero:
- For every x in G, implies
- For every x in G,
- For every x, y in G,
In this case, the function induces on G the structure of a metric space (a type of topology). We can therefore consider G-valued series and define such a series to be absolutely convergent if
In particular, these statements apply using the norm |x| (absolute value) in the space of real numbers or complex numbers.
Read more about this topic: Absolute Convergence
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