Answer:
(0, 16]
Step-by-step explanation:
∑ₙ₌₁°° (-1)ⁿ⁺¹ (x−8)ⁿ / (n 8ⁿ)
According to the ratio test, if we define L such that:
L = lim(n→∞) |aₙ₊₁ / aₙ|
then the series will converge if L < 1.
aₙ = (-1)ⁿ⁺¹ (x−8)ⁿ / (n 8ⁿ)
aₙ₊₁ = (-1)ⁿ⁺² (x−8)ⁿ⁺¹ / ((n+1) 8ⁿ⁺¹)
Plugging into the ratio test:
L = lim(n→∞) | (-1)ⁿ⁺² (x−8)ⁿ⁺¹ / ((n+1) 8ⁿ⁺¹) × n 8ⁿ / ((-1)ⁿ⁺¹ (x−8)ⁿ) |
L = lim(n→∞) | -n (x−8) / (8 (n+1)) |
L = (|x−8| / 8) lim(n→∞) | n / (n+1) |
L = |x−8| / 8
For the series to converge:
L < 1
|x−8| / 8 < 1
|x−8| < 8
-8 < x−8 < 8
0 < x < 16
Now we check the endpoints. If x = 0:
∑ₙ₌₁°° (-1)ⁿ⁺¹ (0−8)ⁿ / (n 8ⁿ)
∑ₙ₌₁°° -(-1)ⁿ (-8)ⁿ / (n 8ⁿ)
∑ₙ₌₁°° -(8)ⁿ / (n 8ⁿ)
∑ₙ₌₁°° -1 / n
This is a harmonic series, and diverges.
If x = 16:
∑ₙ₌₁°° (-1)ⁿ⁺¹ (16−8)ⁿ / (n 8ⁿ)
∑ₙ₌₁°° (-1)ⁿ⁺¹ (8)ⁿ / (n 8ⁿ)
∑ₙ₌₁°° (-1)ⁿ⁺¹ / n
This is an alternating series, and converges.
Therefore, the interval of convergence is:
0 < x ≤ 16
Or, in interval notation, (0, 16].
