How do I find the region of convergence of this infinite series?

Question

I am given the following question : Let f(z) = 1/(1+z) [z is a complex number] (a) Expand f(z) about z=1. (b) Find the region of convergence.

I am able to solve part (a) using Taylor series expansion, but how should I proceed with part (b)? Our professor said that the region of convergence is a circle with radius |z-1|<2. How do I arrive at this?

Answer 1

Because$$\frac1{1+z}=\frac1{2+(z-1)}=\frac12\sum_{n=0}^\infty\frac{(-1)^n}{2^n}(z-1)^n\tag1$$when $\lvert z-1\rvert<2$. Besides, the radius of convergence of $(1)$ is $2$.