Use the Ratio Test to determine whether the series is convergent or divergent. Show all steps

Question

Use the Ratio Test to determine whether the series is convergent or divergent. Show all steps

Answer 1

The ratio test will not tell you the answer.

The series diverges absolutely, by the ratio test...BUT is it conditionally convergent?

But a_n does not approach 0; therefore, the series cannot converge

Answer 2

a_n = (-6)ⁿ / n²

a_n+1 / a_n = [(-6)ⁿ⁺¹ / (n+1)²] [ n² / (-6)ⁿ ] = (-6) [ (n+1)² / n² ] = -6 (1 + 1/n)²

lim_n↔∞ l a_n+1 / a_n l = 6 lim_n→∞ (1 + 1/n)² = (6)(1) = 6 > 1.

By the Ratio Test, the series diverges.