Answer
The series converges for all $x$ in the interval $[-1,1]$.
Work Step by Step
We apply the ratio test
$$ \rho=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\lim _{n \rightarrow \infty} |\frac{ x^{ n+1}/(n+1)^{5} }{x^{ n}/n^{5}}|=|x|\lim _{n \rightarrow \infty} \frac{n^5}{(n+1)^5}=|x| $$
Hence, the series $\Sigma_{n=4}^{\infty} x^{ n}/n^{5}$ converges if and only if $|x|\lt1$. That is, the interval of convergence is $(- 1,1)$.
Now, we check the end points:
At $x=-1$, then
$\Sigma_{n=4}^{\infty} x^{ n}/n^{5}=\Sigma_{n=4}^{\infty} (-1)^{ n}/n^{5}$, converges (Alternating Series Test)
At $x=1$, then
$\Sigma_{n=4}^{\infty} x^{ n}/n^{5}=\Sigma_{n=4}^{\infty} 1/n^{5}$, converges (P- Series Test)
Hence, the series converges for all $x$ in the interval $[-1,1]$.
