On Ahlfors' Complex Analysis, he states that the logarithmic series $$\log(1+z)=z-\frac{z^2}{2}+\frac{z^3}{3}-\frac{z^4}{4}+\frac{z^5}{5}-...$$ centered at the origin must have a radius of $1$ because "if the logarithmic series had a radius of convergence greater than $1$, then $\log(1+z)$ would be bounded for $|z| < 1$" which is a contradiction. He further says "similarly, if the binomial series were convergent in a circle of radius $>1$, the function $(1+z)^\mu$ and all its derivatives wpuld be bounded in $|z| < 1$," which leads to a contradiction whenever $\mu$ is a positive integer.
I do not get why "if the logarithmic series had a radius of convergence greater than $1$, then $\log(1+z)$ would be bounded for $|z| < 1$." I believe he ia using the Maximum Principle But wouldn't using the maximum principle require not only the assumption that the given series converges but also the assumption that it converges to $\log(1+z)$? And similarly, wouldn't it require not only the assumption that the binomial series converges but also the assumption that it converges to $(1+z)^\mu$?
Thank you for your help
If a power series (say centered at $0$) has radius of convergence $R>0$, then it is analytic and thus continuous on the open disk $\{ |z|<R \}$, and in particular is continuous on the closed disk $\{ |z|\le r\}$ for any $0<r<R$. Since that closed disk is compact, this continuous function is automatically bounded on $\{ |z|\le r\}$ and therefore on $\{ |z|<r\}$.