Answer
The series converges for all $x$ in the interval $(-1,1)$.
Work Step by Step
We apply the ratio test:
$$ \rho=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\lim _{n \rightarrow \infty} |\frac{ (n+1)^7x^{n+1} }{n^7x^{n}}|=|x|\lim _{n \rightarrow \infty} \frac{(n+1)^7}{n^7}=|x| $$ hence, the series $\Sigma_{n=8}^{\infty} n^7x^{n}$ 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
$\Sigma_{n=8}^{\infty} n^7x^{n}=\Sigma_{n=8}^{\infty} n^7(-1)^{n}$, diverges (Limit Test)
At $x=1 $, then
$\Sigma_{n=8}^{\infty} n^7x^{n}=\Sigma_{n=8}^{\infty} n^7 $, diverges (Limit Test)
Hence, the series converges for all $x$ in the interval $(-1,1)$.
