Limit Comparison Test

Question

Use the Limit Comparison Test to determine if each series converges or diverges. Do not use the root test.

∑(Infinity)(n=1) ((2n+3)/(5n+4))^n

Answer 1

Let a_n = [(2n + 3)/ (5n + 4)]ⁿ and b_n = [2n / 5n]ⁿ = (2/5)ⁿ.

lim_n→∞ (a_n / b_n) = 1 > 0

∑b_n is a convergent geometric series. So, by the Limit Comparison Test, ∑a_n also converges.