Final answer:
The radius of convergence, R, of the series Σ[(n^8 + 1) / (x - 4)^n] is (5, ∞).
Step-by-step explanation:
To find the radius of convergence, we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms in a series is less than 1, then the series converges.
In this case, the series is Σ[(n^8 + 1) / (x - 4)^n]. Applying the ratio test:
limn→∞ |((n+1)^8 + 1) / (x - 4)^(n+1) / (n^8 + 1) / (x - 4)^n| < 1
limn→∞ |(n+1)^8 + 1| / |n^8 + 1| * |(x - 4)^n / (x - 4)^(n+1)| < 1
limn→∞ |(n+1)^8 + 1| / |n^8 + 1| * 1 / (x - 4) < 1
limn→∞ |(n^8 + 8n^7 + 28n^6 + 56n^5 + 70n^4 + 56n^3 + 28n^2 + 8n + 2) / (n^8 + 1)| * 1 / (x - 4) < 1.
We can simplify the limit as n approaches infinity:
limn→∞ (1 + 8/n + 28/n^2 + 56/n^3 + 70/n^4 + 56/n^5 + 28/n^6 + 8/n^7 + 2/n^8) / 1 * 1 / (x - 4) < 1.
As the limit approaches infinity, all the terms with n in the denominator go to 0:
limn→∞ (1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0) / 1 * 1 / (x - 4) < 1.
So we are left with:
1 / (x - 4) < 1.
We can then solve for x:
(x - 4) > 1.
x > 5.
Therefore, the series converges for all x greater than 5. The interval of convergence is (5, ∞).