Is this function, a sum of one term and a convergent series, analytic?

Question

$$(\frac{1}{z} + \sum z^n)$$

for 0<|z|<1.

This is for complex variables. So, the series, convergent for the above domain of definition, always represents an analytic function.

What about the 1/z term? Of course, it's bounded away from 0, and the inequalities for the domain of definition are strict. So does that make the 1/z term bounded...for sure? Silly question, I know. Sorry.

Thanks,