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I need to find the radius of convergence for:

$$\sum \ln j^3 x^j$$

By the ratio test, I get:

$$\displaystyle\frac{\ln (j+1)^3 x^{j+1}}{\ln j^3x^j}$$

However, I'm not sure what happens to the ln parts in terms of convergence?

kiwifruit
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  • The limit test applied should give:$$\lim_{j\to\infty}\left|\displaystyle\frac{\ln (j+1)^3 x^{j+1}}{\ln j^3x^j}\right|$$Not$$\displaystyle\frac{\ln (j+1)^3 x^{j+1}}{\ln j^3x^j}$$ –  Apr 20 '14 at 17:31
  • Yes, I know, I just wrote out the inside ratio and I am having trouble finding the limit – kiwifruit Apr 20 '14 at 17:33

1 Answers1

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$$\lim_{j\to\infty}\left|\displaystyle\frac{\ln (j+1)^3 x^{j+1}}{\ln j^3x^j}\right|=|x|\lim_{j\to\infty}\left|\displaystyle\frac{\ln (j+1) }{\ln j}\right|=|x|<1$$ Use L'Hopital's for the last step.