Question - Determination of Interval of Convergence for Power Series Options
Solution:
The image shows a question asking which of the provided power series has an interval of convergence of $$0 < x \leq 2$$. We have four power series options (A, B, C, and D) to consider.To determine the interval of convergence, we usually use the ratio test. The ratio test tells us that a series ∑a_n converges if the limit as n goes to infinity of |a_{n+1} / a_n| is less than 1.Let's apply the ratio test to each option:Option A:The general term is $$ a_n = (-1)^n (x - 1)^{n+1} / (n + 1) $$. Using the ratio test,|a_{n+1} / a_n| = |(-1)^{n+1} (x - 1)^{n+2} / (n + 2)| * |(n + 1) / (-1)^n (x - 1)^{n+1}|= |(x - 1)^{n+2} / (n + 2)| * |(n + 1) / (x - 1)^{n+1}|= |(x - 1)| * |(n + 1) / (n + 2)|As n approaches infinity, |(n + 1) / (n + 2)| approaches 1, so we have |x - 1| < 1 for convergence. This results in an interval of convergence -1 < x - 1 < 1, which simplifies to 0 < x < 2. However, we need to check the endpoints separately to see if $$x = 2$$ is included in the interval of convergence. If we substitute $$x = 2$$ into the series, we get an alternating series $$∑ (-1)^n / (n + 1)$$ which converges by the alternating series test. Therefore, option A has an interval of convergence $$0 < x \leq 2$$.Without testing the other options, we already know that option A is the correct answer to the question. This is because the question asked for the series that has an interval of convergence of $$0 < x \leq 2$$. If you need to analyze the other options, please let me know, and I can perform a similar analysis on each of them.
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