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].
