Question

Find the intervals of convergence of f(x), f '(x), f ''(x), and ∫f(x) dx. (Be sure to include a check for convergence at the endpoints of the intervals. Enter your answer using interval notation.) f(x) = [infinity] (−1)n + 1(x − 8)n n8n n = 1

Answer 1

To find the intervals of convergence of a series, we can use the Ratio Test. The series converges when the absolute value of the quotient of consecutive terms is less than 1. The intervals of convergence for this specific series are (-1, 9) and (7, 9).

Step-by-step explanation:

To find the intervals of convergence of the function f(x), we can use the Ratio Test:

Let an = (-1)n+1(x-8)n/(n8n)

Using the Ratio Test, we evaluate the limit as n approaches infinity of |an+1/an|:

|(x-8)/(8)|

The series f(x) converges when |(x-8)/(8)| < 1. Solving this inequality, we find two intervals: (-1, 9) and (7, 9). To check for convergence at the endpoints, we substitute x = -1 and x = 9 into f(x) and ensure the series converges.

Answer 2

Answer: See solution and explanations in the attached documents

Step-by-step explanation:

See explanations in the attached documents