Use the Ratio Test to discuss the convergence or divergence of a series

Question

Use the Ratio Test to discuss the convergence or divergence of the series which is the sum from

n=1 to infinity of the terms a_n=[(-1)ⁿ][n³]/3ⁿ

Answer 1

a_n = (-1)ⁿn³/3ⁿ; a_n+1 = (-1)ⁿ⁺¹(n + 1)³/3ⁿ⁺¹.

lim n→∞ Ia_n+1/a_nI = lim n→∞ I(n + 1)³/(3n³) = 1/3 < 1.

So, given series converges