Explanation:
∑ₙ₌₁°° (2x + 3)ⁿ / (n9ⁿ)
Use ratio test:
lim(n→∞)│aₙ₊₁ / aₙ│< 1
lim(n→∞)│[(2x + 3)ⁿ⁺¹ / ((n+1) 9ⁿ⁺¹)] / [(2x + 3)ⁿ / (n9ⁿ)]│< 1
lim(n→∞)│(2x + 3) n / (9 (n+1))│< 1
│(2x + 3) / 9│< 1
│2x + 3│< 9
-9 < 2x + 3 < 9
-12 < 2x < 6
-6 < x < 3
If x = -6, ∑ₙ₌₁°° (2(-6) + 3)ⁿ / (n9ⁿ) = ∑ₙ₌₁°° (-1)ⁿ / n, which converges.
If x = 3, ∑ₙ₌₁°° (2(3) + 3)ⁿ / (n9ⁿ) = ∑ₙ₌₁°° 1 / n, which diverges.
The interval of convergence is therefore [6, 3).
∑ₙ₌₁°° (x + 4)ⁿ / n!
Use ratio test:
lim(n→∞)│aₙ₊₁ / aₙ│< 1
lim(n→∞)│[(x + 4)ⁿ⁺¹ / (n+1)!] / [(x + 4)ⁿ / n!]│< 1
lim(n→∞)│(x + 4) n! / (n+1)!│< 1
lim(n→∞)│(x + 4) / (n + 1)│< 1
0 < 1
The interval of convergence is therefore (-∞, ∞).