How do you do these two questions?

Step-by-step 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 (-∞, ∞).

a box 3 3/4 feet long or 4+ feet.

step-by-step explanation: