The radius of convergence of a power series tells how far out the series will converge.
Let be an infinite sequence of complex numbers, let be a particular complex number, and consider the power series
The radius of convergence is the supremum of the positive numbers such that (1) converges for . (This supremum is a nonnegative lower real in .)
The phrasing above is ambiguous. If is less than the radius of convergence, does this mean that (1) converges for every with or for some such ? It doesn't matter:
(We do not say that (1) converges for with , but this has no effect on the supremum.)
The radius of convergence is clearly independent of . It can be calculated quite easily from the coefficients :
The radius of converge of (1) is
For , (1) is (pretty much by definition) an analytic function of . There is a partial converse:
If a function is analytic at all with , then there is a power series (1) that converges to for all with .
(Specifically, .)
Over the real numbers, Theorem fails. I'm sure (says one of this pages authors) that there are interesting things to say about series in adic numbers, matrices, and things like that, but I don't know them.
Last revised on July 5, 2013 at 11:36:49. See the history of this page for a list of all contributions to it.