Finding Domain of Convergence for Complex Series

[elimenohpee]

Homework Statement

I need to find the domain of absolute convergence of the following series:

$$^{\infty}_{1}$$$$\sum(z+3)^{2n}/(2n)!$$

Homework Equations

Ratio test?

The Attempt at a Solution

I'm not really sure how to handle the complex variable z within the series. I attempted to use the ratio test and simplified down to this:

$$lim (n->\infty) |(z+3)^{2}/(2n+1)(2n+2)|$$

I'm assuming I simplified this down to this point correctly of course. Can someone nudge me in the right direction? I just need to know how to handle z.

 

[Dick]

z really doesn't need any 'handling'. It's just some fixed complex number. What's the limit as n -> infinity?

 

[elimenohpee]

It would be zero correct?

I can't find any examples with complex series, but from my text it states they are almost identical to real valued series. If the series ended up being non-zero, is the answer just expressed in terms of z?

 

[Chantry]

It would be zero.

(z + 3)^2 = 8 + 6z = 8 + 6(-1)^0.5, which is a constant.

So the limit becomes,

|8+6z| * the limit

And the limit approaches zero.

 

[Dick]

It would be zero correct?

I can't find any examples with complex series, but from my text it states they are almost identical to real valued series. If the series ended up being non-zero, is the answer just expressed in terms of z?

If the limit depends on z, then you need to express it in terms of z. Here it doesn't.