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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.

Use the alternating series test to determine the convergencedivergence of the series The alternating series fails The series converges The series is conditional class=

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MathPhys

Answer:

The series converges.

Step-by-step 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.

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