Let $(a_k)$, with $0<\liminf |a_k|\le\limsup |a_k|<\infty$. Determine the radius of convergence of $\sum a_kX^k$.

I want to check if my reasoning is correct. Because $0<\liminf |a_k|\le\limsup |a_k|<\infty$ then the sequence $(|a_k|)$ is bounded and eventually different than zero.

And we have that for any fixed $\ell$, $\limsup\sqrt[k]{|\ell|}=1$, then exists some $0<m\le|a_k|$ and some $M\ge|a_k|$ for all $k\ge N$, for enough large $N$. Then


Then the radius of convergence is



Eventually, $(|a_n|)$ is bounded below by some $\epsilon >0$ (otherwise $0$ is a cluster point, so $\liminf \le 0$) and above by some $M$ (otherwise $\limsup = \infty$). So we have:

$$\epsilon \sum |x|^n \le \sum |a_n x^n| \le M \sum |x|^n$$

Which shows that the radius of convergence is $1$.


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