Let $$f(w) = \sum_{n\ge0} (w-a)^n$$ for every $w \in B(a,1) = \left\{z \in\mathbb C\, :\, |z − a| < 1\right\}$?
I've written until now the equation of Mobius transformation which is $$w=\frac{az+b}{cz+d}$$ and its inverse $$w'=\frac{ab-bc}{(cz+d)^2}$$
How exactly do I show using the equation that the function is a Mobius Transformation?
Hint: your infinite sum is a geometric series. The sum of a geometric series can be written in closed form.