How do you do these questions?

Step-by-step 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.