Question

How do you do this question?

Answer 1

Answer:

[4, 6]

Step-by-step explanation:

∑ₙ₌₁°° (x − 5)ⁿ / n²

Use ratio test.

lim(n→∞)│aₙ₊₁ / aₙ│< 1

lim(n→∞)│[(x − 5)ⁿ⁺¹ / (n+1)²] / [(x − 5)ⁿ / n²]│< 1

lim(n→∞)│[(x − 5) n² / (n+1)²│< 1

│x − 5│< 1

-1 < x − 5 < 1

4 < x < 6

If x = 4, ∑ₙ₌₁°° (4 − 5)ⁿ / n² = ∑ₙ₌₁°° (-1)ⁿ / n², which converges.

If x = 6, ∑ₙ₌₁°° (6 − 5)ⁿ / n² = ∑ₙ₌₁°° 1 / n², which converges.

So the interval of convergence is [4, 6].