If $n^3 < |a_n| < n^4$ find the radius of convergence for $\sum_{n=2}^\infty a_nx^n$

Question

If $n^3 < |a_n| < n^4$ find the radius of convergence for $\sum_{n=2}^\infty a_nx^n$

Could someone explain how he got inequality (1)? Theorem 4.1 stated that a power series converges if $|x| <R$.

Answer 1

The radius of convergence of the series $\sum_n b_nx^n$ is the supremum of the set $$\{t\geqslant 0, \sup_{n\geqslant 0}|b_n|t^n<+\infty\}.$$ Since $$\{t\geqslant 0, \sup_{n\geqslant 0}n^4t^n<+\infty\}\subset \{t\geqslant 0, \sup_{n\geqslant 0}|a_n|t^n<+\infty\} \subset \{t\geqslant 0, \sup_{n\geqslant 0}n^3t^n<+\infty\},$$ the conclusion follows.