real analysis - Are the limits $limlimits_{nto infty }left|frac{a_{n+1}}{a_n}right|,$ and $limlimits_{nto infty }sqrt[n]{|a_n|},$ equal?

Let consider the power series $$
\sum_{k=1}^\infty a_kx^k.
$$
We know that it converges when
$$\lim_{n\to \infty }\sqrt[n]{|a_n|}|x|< 1\quad\text{or}\quad |x|< \frac{1}{\lim_{n\to \infty }\sqrt[n]{|a_n|}}$$

But D'Alembert test tells us that it converges if $$|x|< \frac{1}{\lim_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|},$$
so since the radius of convergence is unique, we should have that $$\lim_{n\to \infty }\sqrt[n]{|a_n|}=\lim_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|.$$

I tried to prove it, but this result looks strange to me. So if it doesn't work always, how can we have those to limit as radius of convergence? I guess the equality of those two limit should be true most of the time.
So under what conditions is it true?

Answer

If the limit
$$
\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|,

$$
exists in $[0,\infty]$, then so does
$$
\lim_{n\to\infty}\sqrt[n]{|a_n|}
$$
and the limits are equal.

However, if the second limit exists, then the first one DOES NOT HAVE to exist. For example
$$
a_n=\left\{\begin{array}{lll}

1 & \text{if} & n\,\, \text{odd},\\
2 & \text{if} & n\,\, \text{even},
\end{array}\right.
$$
then
$$
\lim_{n\to\infty}\sqrt[n]{|a_n|}=1,
$$
while the first limit does not exist.

In general,
$$
\liminf_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\le
\liminf_{n\to\infty}\sqrt[n]{|a_n|}\le
\limsup_{n\to\infty}\sqrt[n]{|a_n|}\le
\limsup_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| \tag{1}
$$

Note. A consequence of $(1)$ is that, if both limits exist, then they are equal. In fact, if the ratio limit exists, then root limit also exists, and their are equal. However, it is possible for the root limit to exist, while the ratio one to not exist.