Prove that the function is analytic when and that its derivative is . Then demonstrate that the derivative of is worth and deduce that . Note that .

My attempt: where , , ,... . So it seems to me that f is analytic for any value of z though I don't know how to formally prove it. So I don't understand why they bother with |z|<1.

I'd like to tackle this exercise and eventually finish it with your help. Thanks for any comment/tip.