Answer
Conditionally convergent
Work Step by Step
Alternating series Test: Let us suppose a series $a_n$ such that $a_n=(-1)^{n+1} b_n$ or $a_n=(-1)^n b_n$ when $b_n \geq 0$ for all $n$.
1) $\lim\limits_{n \to \infty}b_n=0$
2) $b_n$ shows a decreasing sequence.
For a series to be convergent, the above two conditions must be satisfied.
We have: $b_k=\dfrac{k^2}{k^3+1}$
1) We see that $\lim\limits_{k \to \infty}b_k=\lim\limits_{k \to \infty} \dfrac{k^2}{k^3+1}\\=\dfrac{2}{6k}\\=0$
(2) In $b_k$, the denominator is increasing, so $b_k$ shows a decreasing sequence.
We conclude that both conditions of the Alternating series Test are satisfied, so the given series is conditionally convergent because the absolute value of the function diverges (harmonic series).
