## Calculus (3rd Edition)

The radius of converges is $R=1$ and the series converges at $-1\leq x \leq 1$.
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.