Complex Analysis: Radius of Convergence

[gbean]

Homework Statement

Find the radius of convergence of the power series:
a) [tex]\sum[/tex] z[tex]^{n!}[/tex]
n=0 to infinity

b) [tex]\sum[/tex] (n+2[tex]^{n}[/tex])z[tex]^{n}[/tex]
n=0 to infinity

Homework Equations

Radius = 1/(limsup n=>infinity |cn|^1/n)

The Attempt at a Solution

a) Is cn in this case just 1? And plugging it in, the radius is 1?

b) cn = n+2[tex]^{n}[/tex], so then limsup n=> infinity |n+2[tex]^{n}[/tex]|[tex]^{1/n}[/tex] => ?? I'm stuck at this point.

i'm also confused in general, is cn just a sequence of coefficients, and what is zn? And I have other formulas for figuring out the radius of convergence, such as the ratio test. I'm not sure when to use which methods. Thank you!

 

[gbean]

I also don't understand why z^n isn't used in the calculation of the radius of convergence.

 

[gbean]

So I'm running into trouble for part b still, any help would be greatly appreciated. The answer key says 1/2, but I don't know how to derive that.