In class today we were given the power series
$\sum_{n=2}^{\infty} \frac{1}{n^22^{3n-2}}(x-3)^{3n+1}$
and we have to find the radius of convergence
After running through all the working I get to
$|x-3|^3 < 8$
Does this mean that the radius of convergence is 8 or would it be $\sqrt[3]{8}$ I am just a little confused
The radius of convergence is $2$ and then the given series converges for $|x-3|<2$, then we need to check directly the cases $x-3=\pm2 $, that is
$$\sum_{n=2}^{\infty} \frac{(-2)^{3n+1}}{n^22^{3n-2}},\quad \sum_{n=2}^{\infty} \frac{(2)^{3n+1}}{n^22^{3n-2}}$$