🏗️ $\Theta \rho ϵ\eta \Pi \alpha \tau \pi$🚧 (under construction)

Given a power series ${\sum }_{n=0}^{\infty }{a}_n{x}^n$ then either there exists a set $C$ such that the series converges for any $x\in C$ and diverges otherwise. $C$ is either of the form $[-M,M]$ or $\mathbb{R}$ itself, such that

• For each $[a,b]\subseteq C$ the series converges uniformly
• If $\alpha ={lim sup}_{n\to \infty }{\left|a_n\right|}^{\frac{1}{n}}$, then

$${\displaystyle C=\{\begin{array}{cc}\mathbb{R}\hfill & \text{ if }\alpha =0\hfill \\ \varnothing \hfill & \text{ if }\alpha =+\infty \hfill \\ [-\frac{1}{\alpha },\frac{1}{\alpha }]\hfill & \text{ if }\alpha \in {\mathbb{R}}^{+}\hfill \end{array}}$$