Answer
Diverges.
Work Step by Step
Plugging $n = {1,2,3,4,5}$ in $\dfrac{{(-1)^{n}2n^3}}{{n^3+1}}$.
$\implies n=1$: $\dfrac{{(-1)^{1}2(1)^3}}{{1^3+1}} =-1$
$\implies n=2$: $\dfrac{{(-1)^{2}2(2)^3}}{{2^3+1}} =1.7777778$
$\implies n=3$: $\dfrac{{(-1)^{3}2(3)^3}}{{3^3+1}} =-1.9285714$
$\implies n=4$: $\dfrac{{(-1)^{4}2(4)^3}}{{4^3+1}} =1.9692308$
$\implies n=5$: $\dfrac{{(-1)^{5}2(5)^3}}{{5^3+1}} =-1.984127$
We see that $\lim\limits_{n \to \infty}a_n\ne \lim\limits_{n \to \infty}a_{n+1}$
Therefore, the given series diverges.
