Rules for determining interval of convergence of power series

Statement

Consider a power series of the form:

$\sum _{k=0}^{\mathrm{\infty }}{a}_k{x}^k$

The rules here help determine the radius of convergence and interval of convergence for this power series in terms of the manner in which the coefficients $a_k$ grow or decay with $k$ as $k\to \mathrm{\infty }$.

Below is a table with all the cases:

Verbal description of case	Lim sup version for coefficients	Other version for coefficients	Conclusion about radius of convergence	Conclusion about interval of convergence	Examples of functions $k\mapsto a_k$
coefficients decay superexponentially	$lim{sup}_{k\to \mathrm{\infty }}|a_k{|}^{1/k}=0$	[SHOW MORE] For any $0<\lambda <1$, there exists a positive integer $k_0$ such that $\left|a_k\right|<{\lambda }^k\mathrm{\forall }k\ge k_0$	$\mathrm{\infty }$	all of $\mathbb{R}$	reciprocal of function with doubly exponential growth, reciprocal of exponential of function with polynomial growth, reciprocal of factorial function or factorial-type function.
coefficients grow superexponentially	$lim{sup}_{k\to \mathrm{\infty }}|a_k{|}^{1/k}=\mathrm{\infty }$	Hard (see details below)	0	$\{0\}$	function with doubly exponential growth, exponential of function with polynomial growth, factorial or factorial-type function
coefficients grow or decay exponentially	$lim{sup}_{k\to \mathrm{\infty }}|a_k{|}^{1/k}=\lambda$	$1/\lambda$	To determine whether endpoints are included, we use the degree difference test or some variant (see below)	exponential function, exponential function times rational function times function of slower growth