I am self studying concepts of complex analysis from Ponnusamy and Silvermann "Complex Variables with Applications"
In the chapter of analytic continuation in basic concepts authors mention that the power series $f(z) = \sum\limits_{n=0 }^ {\infty } z^n $ converges for $|z|<1$, and hence $f(z)$ is analytic in the disk $|z|<1$ and represents the function $f(z) =\frac1{(1-z)}$.
I have doubt in this line which follows just after this paragraph- Although power series diverges at each point on $|z|=1$, $f(z)$ is analytic in whole of complex plane except $1$.
As power series converges in $|z|<1$, so the function $f(z)$ is analytic in $|z|<1$. But my doubt is how $f(z)$ is analytic for whole of complex plane except $1$. The limit of derivative would be unbounded if $z>1$. Also $f(z) \neq \frac1{(1-z)}$. So, I cannot differenciate $\frac1{1-z}$ in any neighborhood of point I want to integrate if point is $|z|>1$.
So how $f(z)$ is analytic for whole complex plane except $1$.
Can someone please explain?