Are the limits $\lim\limits_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|\,$ and $\lim\limits_{n\to \infty }\sqrt[n]{|a_n|}\,$ equal?

Solution 1:

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.