Determination of Interval of Convergence for Power Series Options
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.
