Explanation:
∑ₙ₌₀°° aₙ xⁿ has a radius of convergence of 2, which means│x│< 2.
∑ₙ₌₀°° aₙ 3ⁿ/2ⁿ means x = 3/2. Since │3/2│< 2, the series converges.
i) ∑ₙ₌₀°° bₙ xⁿ converges when x = -4.
∑ₙ₌₀°° bₙ (-4)ⁿ = ∑ₙ₌₀°° (-1)ⁿ bₙ (4)ⁿ. This is an alternating series. So it converges if lim(n→∞) bₙ (4)ⁿ = 0. Which means lim(n→∞) bₙ < (¼)ⁿ.
Therefore bₙ (4)ⁿ < (r)ⁿ where r < 1, and ∑ₙ₌₀°° bₙ (4)ⁿ converges by comparison test.
ii) From (i), we found lim(n→∞) bₙ < (¼)ⁿ. Therefore, ∑ₙ₌₀°° bₙ converges by comparison test.
ii) ∑ₙ₌₀°° (-1)ⁿ bₙ (9)ⁿ = ∑ₙ₌₀°° bₙ (-9)ⁿ, so x = -9.
However, we do not know if -9 is within the interval of convergence. So there is not enough information.