radius of convergence
To the power series
there exists a number , its radius of convergence, such that the series converges absolutely for all (real or complex) numbers with and diverges whenever . This is known as Abel’s theorem on power series. (For no general statements can be made.)
The radius of convergence is given by:
and can also be computed as
if this limit exists.
It follows from the Weierstrass -test (http://planetmath.org/WeierstrassMTest) that for any radius smaller than the radius of convergence, the power series converges uniformly within the closed disk of radius .
|Title||radius of convergence|
|Date of creation||2013-03-22 12:32:59|
|Last modified on||2013-03-22 12:32:59|
|Last modified by||PrimeFan (13766)|
|Synonym||Abel’s theorem on power series|