Question

Find the radius of convergence of the power series \sum_{n=1}^\infty \frac{x^n}{\root 9 \of n}

Answer 1

Assuming "root 9 of n" is supposed to mean "the ninth root of n", that is $\sqrt[9]n$ we can use the ratio test, which says the series converges whenever

$\displaystyle\lim_{n\to\infty}\left|\frac{\frac{x^{n+1}}{\sqrt[9]{n+1}}}{\frac{x^n}{\sqrt[9]n}}\right|$

We have

$\displaystyle|x|\lim_{n\to\infty}\frac{\sqrt[9]n}{\sqrt[9]{n+1}}=|x|\sqrt[9]{\lim_{n\to\infty}\frac n{n+1}}=|x|$

which means the radius of convergence for this power series is 1.