real analysis - Applying root test to sequence $frac{1}{2} + frac{1}{3} + frac{1}{2^2} + frac{1}{3^2} + cdots$

The following is an example from Principles of Mathematics, by Rudin. I've been trying to understand the example but haven't quite grasped it because it seems I can solve it differently.

Given the following sequence: $\displaystyle \frac{1}{2} + \frac{1}{3} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{2^3} + \frac{1}{3^3} + \cdots$

Using the Ratio Test:

$$\lim \inf_{n \to \infty} \frac{a_{n+1}}{a_{n}} = \lim_{n \to \infty} (\frac{2}{3})^n = 0$$

$$\lim \sup_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{1}{2}(\frac{3}{2})^n = +\infty$$

Using the Root Test:

$$\lim \inf_{n \to \infty} (a_n)^{\frac{1}{n}} = \lim_{n \to \infty} (\frac{1}{3^n})^{\frac{1}{2n}} = \frac{1}{\sqrt{3}}$$

$$\lim \sup_{n \to \infty} (a_{n})^{\frac{1}{n}} = \lim_{n \to \infty} (\frac{1}{2^n})^{\frac{1}{2n}} = \frac{1}{\sqrt{2}}$$

What I don't understand is how to find the $\lim \sup$ and $\lim \inf$ for the ratio test. I also don't understand why for the root test, we are looking at the $2n^\text{th}$ root. Where does this 2 come from? Furthermore, are we looking at $a_n$ as alternating between $\frac{1}{2^m}$ and $\frac{1}{3^m}$ or is $a_{n}$ actually $\frac{1}{2^m} + \frac{1}{3^m}$?

As a side note, I do know how to solve this question if asked whether or not this series converges. I simply don't understand the book went around solving it.