= Radius of convergence of a sparse power series
{title2=$R^{-1}=\limsup_n|a_n|^{1/m_n}$}
For a series $\sum_n a_n x^{m_n}$ with distinct increasing integer exponents, define the ordinary coefficients to be $a_n$ at $m_n$ and zero elsewhere. The <Cauchy-Hadamard theorem> then gives
$$
R^{-1}=\limsup_n|a_n|^{1/m_n}.
$$
The exponent in the root is the actual degree $m_n$, not the index $n$. For $a_n=(n!)^2$ and $m_n=n^2$, $0\leq2\log(n!)/n^2\leq2\log n/n\to0$, so $R=1$. Boundary behavior is a separate question.
Back to article page
