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.
Oh thanks a lot. I will try to do the algebra on my own. So you applied the root test and saw that the radius of convergence depends on |z|. And if |z|<1 then the series converges absolutely and the function is therefore analytic.
Is there a theorem that states that any absolutely convergent power series is an analytic function? I personally don't find it "that obvious".
Correct me if I'm wrong: Any analytic function can be written as an infinite series. Must these infinite series converge absolutely? Or maybe a simple convergence is enough for some series to be analytic functions?