# Power series and ratio test

When we are dealing with finding radius of convergence for a power series , we apply the ratio test and it works . I'm curious about the case in which ratio test isn't helpful and we should use another tests . I've tried many examples but didn't get any proper result .

Counterexample $$\frac 12 + \frac 13x + \frac 1{2^2}x^2 +\frac 1{3^2} x^3 + \frac 1{2^3} x^4 + \frac 1{3^3}x^5 + \cdots,$$ then the ratio test does not work, but the Cauchy root test could confirm.
Alternate form: $$\sum_0^{+\infty} a_n x^n$$, where $$a_n = \begin{cases} \dfrac 1{2^{k+1}}&n = 2k, \\ \dfrac 1{3^{k+1}} & n =2k+1, \end{cases} \quad [k \in \mathbb N].$$