Question

Use the alternating series test to determine the convergence/divergence of the series

The alternating series fails.
The series converges.
The series is conditionally convergent.
The summation must start with n = 0 so the alternating series test cannot be applied.

Answer 1

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.