Final answer:
The interval of convergence for the power series is found using the ratio test and by checking the endpoints of the interval. After applying the test and checking endpoints, the interval of convergence is determined.
Step-by-step explanation:
The student's question revolves around determining the interval of convergence for the given power series ∑ₙ₁ (-1)n (x-2)n / n3. This requires using the ratio test, where we take the limit as n approaches infinity of the absolute value of the n+1-th term over the n-th term of the series and set this less than 1 to find the values x for which the series converges.
To apply the ratio test, we would calculate: limn → ∞ |((-1)n+1 (x-2)n+1 / (n+1)3) / ((-1)n (x-2)n / n3)| = |(x-2) / (n+1)|3 * n3. As n approaches infinity, this limit simplifies to |x-2| and we set this less than 1 to find the interval of convergence. We solve the inequality |x-2| < 1 to get the interval (1, 3). However, we must also check the endpoints, x=1 and x=3, by plugging them into the original series and testing for convergence at these points.
Finally, after checking the endpoints, the interval of convergence would be the set of all x for which the original series converges. This interval could be either open or closed at the endpoints depending on whether the series converges or diverges when x equals the endpoints.