The radius of convergence for the power series : $\ln x+\ln^{2}x + \ln^{3}x +\dots$

Question

I was tasked with finding the radius of convergence for the power series : $$\ln x+\ln^{2}x + \ln^{3}x +\dots$$ the problem here lies in the fact that $\ln^{n}x$ has no $a_{n}$ part and so neither the d'Alembert criteria nor Cauchy can be applied. Also plugging $a_{n}=1$ is useless. How can this be evaluated? Should it be converted to Taylor, or integrated?

Answer 1

$$\ln x+\ln^2x+...=\sum_{k=0}^{\infty}(\ln x)^k-1=\frac{1}{1-\ln x}-1$$ provided that $|\ln x|<1$ or equivalently $e^{-1}<x<e$. This follows from the geometric series.