Answer
The series $\sum_{n=1}^{\infty} \frac{(-1)^{n}n^4}{ n^3+1}$ diverges
Work Step by Step
We have the absolute series
$$\sum_{n=1}^{\infty} |\frac{(-1)^{n}n^4}{ n^3+1}|=\sum_{n=1}^{\infty} \frac{n^4}{ n^3+1}.$$
So, we have $$\lim_{n\to \infty }b_n=\lim_{n\to \infty } \frac{n^4}{ n^3+1}=\lim_{n\to \infty } \frac{n }{ (1/n)+(1/n^4)}=\infty\neq 0.$$
Then the positive series diverges by the divergence test.
The terms $\frac{n^4}{ n^3+1}$ do not form a decreasing sequence. Thus, the series $\sum_{n=1}^{\infty} \frac{(-1)^{n}n^4}{ n^3+1}$ does not converge conditionally or absolutely.
