Laurent and Taylor series in the unit disc

[Grothard]

Homework Statement

Let f(z) be a function that is analytic for all |z|≤1, with the exception of z_0, which lies on the circle |z|=1. f(z) has a first order pole at z_0. Letting Ʃ a_n z^n be the Maclaurin expansion of the function, prove that z_0 = lim_(n→∞) a_n/a_(n+1)

Homework Equations

Taylor expansion about z=0, Laurent expansion about z=z_0, Residue theorems

The Attempt at a Solution

I have studied this one extensively but can't quite see where that resulting limit is coming from. The fact that there is overlap between the regions of convergence of the Taylor and the Laurent series I think might be key, as is means we can fix a value of z that works in both series. I tried setting the Taylor series equal to the Laurent series about z_0, but to extract all the z_0 out of terms in the form b_n (z-z_0)^n I would have to use a lot of the binomial theorem, which doesn't seem to go anywhere.

 

[lippyka]

I also tried using the residue theorem, but that just yields an expression for a_0 (the residue at z_0). Any help is much appreciated.