Question

How do you do this question?

Answer 1

Divergent because if n is equal to one that makes 1^2 • 5^1-1 / (-6) ^1 to 2•5/-6 to 10/-6 which estimates to -1.66

Answer 2

Convergent

Explanation:

aₙ = n² 5ⁿ⁻¹ / (-6)ⁿ

aₙ = (-1)ⁿ n² 5ⁿ⁻¹ / 6ⁿ

aₙ = (-1)ⁿ n² 5ⁿ / (5 (6ⁿ))

aₙ = ⅕ (-1)ⁿ n² (5/6)ⁿ

bₙ = ⅕ n² (5/6)ⁿ

bₙ = ⅕ n² / (6/5)ⁿ

lim(n→∞) bₙ

lim(n→∞) [⅕ n² / (6/5)ⁿ]

Use L'Hopitals' rule:

lim(n→∞) [⅖ n / ((6/5)ⁿ ln(6/5))]

Use L'Hopitals' rule again:

lim(n→∞) [⅖ / ((6/5)ⁿ ln(6/5)²)]

0

The limit is 0, so the series converges.