If we studied this as a power series, its properties would include, for example, that its radius of convergence is 1.
22.
Using the ratio test it is possible to show that this power series has an infinite radius of convergence, and so defines for all complex.
23.
The first case is theoretical : when you know all the coefficients c _ n then you take certain limits and find the precise radius of convergence.
24.
Since this holds true for all " x " in the radius of convergence of the original Taylor series, we can compute as follows.
25.
An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity.
26.
Likewise, analytic continuation of a function from the reals to the complex plane yields unique results, especially when the radius of convergence is infinite as here.
27.
The number " r " is called the "'radius of convergence "'of the power series; in general it is given as
28.
So the radius of convergence of any probability generating function must be at least 1, by Abel's theorem for power series with non-negative coefficients.
29.
I imagine that what is needed is a formal treatment of the theory of the " radius of convergence " for BCH . I've never seen such.
30.
Therefore, the function has a unique power series which converges to the function for all complex numbers, i . e ., the radius of convergence is infinity.
