Question

Use the Ratio Test to determine whether the series is convergent or divergent

ₙ₌₁Σ[infinity] (-1)ⁿ⁻¹ (5^n/2^n n³)

Answer 1

To determine whether the series is convergent or divergent, we can use the Ratio Test.

Step-by-step explanation:

To determine whether the series ∑n = 1∞ (-1)n-1 (5n/2n n³) is convergent or divergent, we can use the Ratio Test.

The Ratio Test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, the series converges. If the limit is greater than 1 or does not exist, the series diverges.

Applying the Ratio Test to the given series:

limn → ∞ |((-1)n-1 (5n+1/2n+1 (n+1)³)) / ((-1)n-1 (5n/2n n³))|

simplifies to:

limn → ∞ |(-5/2) (n+1)³ / (n³)|

which further simplifies to:

limn →∞ |(-5/2) (1+1/n)³|

Since the limit is |-5/2 * 1| = 5/2, which is greater than 1, the series diverges.