Why does a power series converge absolutely within the radius of convergence using the root test?

1. Given a series $f(x) = \sum_n c_n (x-x_0)^n$, we define the "radius of convergence" as $R = \frac1{\limsup |c_n|^{1/n}}$. With no proof of the wanted property of $R$, this is just a name.

2. The root test says that $\sum_n a_n $ converges absolutely if $\limsup |a_n|^{1/n} < 1$.

3. Applied to our series $f(x)$, we get absolute convergence at $x$ if $$ \limsup_{n\to\infty} |c_n|^{1/n} |x-x_0| < 1.$$ It is easy to check that this is true if $|x-x_0|<R$.

$\lim \sup |c_n (R-\epsilon)^{n}|^{1/n}=\frac {R-\epsilon} R$. Since $\frac {R-\epsilon} R<1$ the conclusion follows.