Finding the radius of convergence of a series.

[FLms]

Homework Statement

What is the radius of convergence of the Taylor Series of the function [itex]f(z) = z cot(z)[/itex], at the point [itex]z = 0[/itex]?

Homework Equations

Taylor series is given by:
[tex]\sum_{k=0}^{\infty} \frac{f^{(k)}(z_{0})}{k!} (z - z_{0})[/tex]

And the radius R by:
[tex]\lim_{n \to \infty} |\frac{a_{n}}{a_{n+1}}|[/tex]

The Attempt at a Solution

The problem here is to find a pattern to represent the function as a series.
I did some derivatives and tried to substitute [itex]cot(z)[/itex] for it's representation as an exponential, but all I've got is division by zero.

[tex]f(z) = z cot(z) = z (\frac{e^{2z} + 1}{e^{2z} -1})[/tex]

[tex]\frac{d f}{dz} = \frac{-e^{4z} + 1 + 4 z e^{2*z}}{(e^{2z}-1)^2}[/tex]
And so on...
At the point z = 0, the function blows up.

As a trigonometrical representation, it's the same thing.
[tex]\frac{df(z)}{dz} = cot(z) - z csc(z)[/tex]


I'm not really going anywhere here.
Please help.

 

[dikmikkel]

Maybe the convergence radius is zero ?