I was wondering if anyone could offer any tips or hints as to how to tackle this question, I've tried looking around on here and in my notes but I can't seem to find anything.
Consider the sequence
$$ {a_n=\frac{x^n}{n^3}}. $$
B1. Apply the ratio test to find out for which values of $x$ the sequence $(a_n)$ converges.
B2. For which value(s) of $x$ does the ratio test provide no information?
B3. Does $(a_n)$ converge or diverge for these values? Justify your answers.
B4. Give an example of a sequence that converges if $-1<x<1$ and diverges otherwise. Justify your answer.
By ratio test we obtain
$$\left|\frac{x^{n+1}}{(n+1)^3}\frac{n^3}{x^n}\right|=|x|\left(\frac n{n+1}\right)^3 \to |x|$$
then for convergence we need $|x|<1$, then study separately the cases $|x|=1$.