Answer
The radius of converges is $R=1$ and the series converges at $-1\leq x \leq 1$.
Work Step by Step
Apply the ratio test:
\begin{aligned}
\rho &=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\
&=\lim _{n \rightarrow \infty}\left|\frac{x^{n+1}}{(n+1)^{4}+2} \frac{n^{4}+2}{x^{n}} \right| \\
&= |x|\lim _{n \rightarrow \infty}\left|\frac{n^{4}+2}{(n+1)^{4}+2} \right| \\
&= |x|
\end{aligned}
Then the radius of converges is $R=1$ and the series converges for $-1\lt x\lt 1$.
At $x=1$ $\sum _{n=0}^{\infty}\frac{1}{n^4+1}$ converges by the comparison test and at $x= -1$ $\sum _{n=0}^{\infty}\frac{(-1)^n}{n^4+1}$ also converges.
