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}}}{(x^n)/(\sqrt[9]n)}\right|<1`

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|<1`

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