# Finding the radius of convergence of a series.

1. May 1, 2012

### FLms

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

$$\lim_{n \to \infty} |\frac{a_{n}}{a_{n+1}}|$$

3. 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 $cot(z)$ for it's representation as an exponential, but all I've got is division by zero.

$$f(z) = z cot(z) = z (\frac{e^{2z} + 1}{e^{2z} -1})$$

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

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

I'm not really going anywhere here.