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

Mathematics

1 answer:

Svetach [21]3 years ago

7 0

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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