limit comparison test

Question

use the limit comparison test to establish the convergence or divergence of the series which is the sum from n=1 to infinity of the terms a_n = (2n²+3n)/[(5+n⁵)^1/2]

Answer 1

We compare given series with divergent series with a_n = 1/n^1/2(p-series with p = 1/2 < 1)

lim n→∞ I(2n² + 3n)/[(5 + n⁵)^1/2]·1/n^1/2I = lim n→∞ (2n^5/2 + 3n^3/2)/(5 + n⁵)^1/2 = lim n→∞(2 + 3/n)/(5/n⁵ + 1)^1/2 = 2. So, given series diverges