Solution
= Solution
Apply the <Cauchy-Hadamard theorem> to the <power series> coefficients $a_n=[n/(n+\alpha)]^{n^2}$ for $n\geq1$:
$$
|a_n|^{1/n}=\left(1+\frac\alpha n\right)^{-n}\longrightarrow e^{-\alpha}.
$$
For example, taking logarithms gives $-n\log(1+\alpha/n)\to-\alpha$. Therefore
$$
\boxed{R=e^\alpha.}
$$
Any convention for the isolated $n=0$ coefficient has no effect on the <radius of convergence>.