Taylor series for the following

[TheFerruccio]

I have a couple of general questions, combined with this one specific question

Homework Statement

Find the Taylor or MacLauren series centered about the given value for the following function, determine the radius of convergence

Homework Equations

$$\mathrm{Ln}\ z$$, 2

The Attempt at a Solution

I know that
$$\mathrm{Log}\ z = \ln{|z|} + i\left(\mathrm{Arg}\ z\right)$$

But, I don't know where to go from here, nor do I know how to find a relevant pattern for find the radius of convergence. It might just be an issue of not remembering this from Calc 2.

Also, from earlier questions, if have some series that's in terms of something like $$\left(2z+i\right)^{2n}$$ Can I just simplify it to be in terms of $$2^{2n} \left(z+\frac{i}{2}\right)^{2n}$$, then have the radius be the square root of whatever the radius would be if it was in terms of $$\left(z+\frac{i}{2}\right)^{n}$$? Then, if the radius is less than 1/2, then it can be determined that the series about the point $$z+\frac{i}{2}$$ does not converge?

 

[TheFerruccio]

I think, all I have to do is just treat the principle natural log as a regular natural log, not splitting it up into anything the definition calls for, then just evaluate it that way, with the radius of convergence being from the point of evaluation to where the natural log ceases to exist. In this case, by observation, R would be 2, but I'll verify it with a Taylor series.