Find the function f(z) = 1/z given in a Taylor series around 1

Find the function f(z) = 1/z given in a Taylor series around 1. Specify the maximum disk where this representation is valid
Thanks for your help

1/z = 1/(1 + (z - 1))
= 1/(1 - -(z - 1))
= Σ(n = 0 to ∞) (-(z - 1))^n, via geometric series
= Σ(n = 0 to ∞) (-1)^n * (z-1)^n.

This converges when |-(z - 1)| < 1 <==> |z - 1| < 1.
So, the maximal radius R = 1.

I hope this helps!