In Rudins Principles of Mathematical Analysis he says consider the following series
$\frac 12 + \frac 13 + \frac 1{2^2} + \frac 1{3^2} + \frac 1{2^3} + \frac 1{3^3} + \frac 1{2^4} + \frac 1{3^4} + \cdots$
for which
$\liminf \limits_{n \to \infty} \dfrac{a_{n+1}}{a_n} = \lim \limits_{n \to \infty} \left( \dfrac {2}{3} \right)^n =0, $
$\liminf \limits_{n \to \infty} \sqrt[n]{a_n} = \lim \limits_{n \to \infty} \sqrt[2n]{\dfrac{1}{3^n}} = \dfrac{1}{\sqrt{3}}, $
$\limsup \limits_{n \to \infty} \sqrt[n]{a_n} = \lim \limits_{n \to \infty} \sqrt[2n]{\dfrac{1}{2^n}} = \dfrac{1}{\sqrt{2}}, $
$\limsup \limits_{n \to \infty} \dfrac{a_{n+1}}{a_n} = \lim \limits_{n \to \infty} \dfrac 12\left( \dfrac {3}{2} \right)^n =+\infty, $
The root test indicates convergence; the ratio test does not apply.
In the book he defines the root and ratios test for the lim sup. I am not exactly sure how he goes from the lim sup to the lim and also why there is a $2n$ (which I assume comes from even terms of the sequence) in the root test. Also why is he checking the lim inf? I believe that my understanding of lim sups and infs are not well developed or I would probably understand what’s going on.
Also how does he get the terms that he is taking the limit of. A nudge in the right direction to figure this out would be much appreciated. Thank you!!