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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 .

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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 $$a_n = \begin{cases} \dfrac 1{2^{k+1}}&n = 2k, \\ \dfrac 1{3^{k+1}} & n =2k+1, \end{cases} \quad [k \in \mathbb N]. $$

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  • $\begingroup$ Can you provide the closed-form please ? $\endgroup$ – S.H.W Oct 25 '18 at 16:00
  • $\begingroup$ I tried to apply root test but didn't get result . Can you help please ? $\endgroup$ – S.H.W Oct 25 '18 at 16:12

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