Answer:
The series converges.
Explanation:
According to the alternating series test:
For a series ∑(-1)ⁿ aₙ or ∑(-1)ⁿ⁺¹ aₙ
If lim(n→∞) aₙ = 0
and aₙ is decreasing,
then the series converges.
aₙ = n² / (n³ + 1)
Since the power of the numerator is less than the power of the denominator, lim(n→∞) aₙ = 0.
Since n² / (n³ + 1) > (n+1)² / ((n+1)³ + 1), the series is decreasing. (We could also prove this by showing that the derivative is negative.)
Therefore, the series converges.