For a power series $\sum_{j=0}^{\infty}a_j(z-z_0)^j$ the Cauchy-Hadamard formula states that:$$R=\frac{1}{\operatorname{lim sup}\sqrt[n]{|a_n|}}$$ Where $\sqrt[n]{|a_n|}$ is a sequence formed from the set of coefficients $\{a_j\}$ and R is the radius of convergence (ROC).
When I'm trying to apply this to find the ROC for $\sum_{j=0}^{\infty}2^jz^{j^2}$ i get the following: $$\{a_j\}=1,2,4,... \Rightarrow \sqrt[n]{|a_n|} = 1^{\frac10},2^{\frac11},4^{\frac12}...=1,2,2...$$ From which i deduce that $R=\frac12$ but the book says the answer is $R=1$.
What am I doing wrong?
I'm also curious as to why $\sum_{j=0}^{\infty}2^jz^{j^2}$ can be called a power series.
The book I'm using is Fundamentals of Complex Analysis by Saff & Snider, section 5.4 exercise 3(b).