Prove that $\sum_{k=1}^\infty a_k x^k$ converges absolutely for all $|x|<1/a$ if $a\ne0$ and for all $x\in\mathbb R$ if $a=0$.

Question

I am new to this site so I am not well versed on how things are done here. I would like some help with checking my attempt at this proof. What am I missing? Please point me in the right direction. Thank you.

Proof: Let $a_k≥0$ and $a_k^{1/k}→a$ as $k→∞$. By the Root Test,

$\lim_{k→∞}⁡\text{Sup}|a_k |^{1/k}⇒\lim_{k→∞}\text{Sup}|a_k x^k |^{1/k}$

$\lim_{k→∞}⁡\text{Sup}|a_k x^k |^{1/k}$=

|x|$\lim_{k→∞}⁡\text{Sup}|a^{1/k}_k|=a|x|$

If $a≠0$ then it follows that the series converges absolutely when $a|x|<1$ or, $|x|<1/a$. For all $x∈\mathbb{R}$ if $a=0$, then the limit is zero, and by the Root Test the series converges absolutely.