Question

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

∑[n=1 to [infinity]] (nπⁿ) / ((-6)ⁿ⁻¹

Identify aₙ.

Answer 1

Using the Ratio Test, the series ∑[n=1 to ∞] (nπⁿ) / ((-6)ⁿ⁻¹) is convergent because the limit of the ratio of consecutive terms is 0, which is less than 1.

Step-by-step explanation:

To determine whether the series ∑[n=1 to ∞] (nπⁿ) / ((-6)ⁿ⁻¹) is convergent or divergent using the Ratio Test, first identify the general term of the series, which is an = (nπⁿ) / ((-6)ⁿ⁻¹). The Ratio Test states that for a series ∑an, if the limit L = lim (n → ∞) |an+1/an| exists and L < 1, the series converges; if L > 1 or L is infinite, the series diverges; if L = 1, the test is inconclusive.

Let's calculate the limit for our series:

1. Find the next term an+1 which is ((n+1)πn+1) / ((-6)n).
2. Form the ratio an+1/an = [((n+1)πn+1)/((-6)n)] / [(nπⁿ) / ((-6)ⁿ⁻¹)].
3. Simplify the ratio and take the limit lim (n → ∞).

This results in L = lim (n → ∞) (n+1)/(-6n) π. Since π is a constant, we look at the coefficient of n, which tends to -1/6 as n approaches infinity. This means the limit L = 0, which is less than 1. Therefore, the series converges.