Answer:
R = 1
I = [-1, 1)
Explanation:
Use the ratio test:
lim(n→∞)│aₙ₊₁ / aₙ│
lim(n→∞)│[xⁿ⁺¹ / (8(n+1) − 1)] / [xⁿ / (8n − 1)]│
lim(n→∞)│[xⁿ⁺¹ / (8n + 7)] × [(8n − 1) / xⁿ]│
lim(n→∞)│[x (8n − 1) / (8n + 7)]│
lim(n→∞)│[(8n − 1) / (8n + 7)]││x│
│x│
The series converges if the limit is less than 1.
│x│< 1
So the radius of convergence is 1.
Solving for x:
-1 < x < 1
Check the endpoints.
If x = -1, the series is ∑ (-1)ⁿ / (8n − 1), which converges.
If x = 1, the series is ∑ 1 / (8n − 1), which diverges.
So the interval of convergence is [-1, 1).