Question - Determining Interval of Convergence for Power Series
Solution:
The image shows a multiple-choice question number 89 that asks which of the following power series has an interval of convergence of $$0 < x \leq 2$$.There are four options: (A), (B), (C), (D)Each of the options shows a sum of terms in a power series with summation notation, starting from $$n = 0$$ to $$n = \infty$$.To solve this question, we need to find out which series converges on the interval $$0 < x \leq 2$$. Usually, to determine the interval of convergence of a power series, we use the ratio test or the root test. However, as this is a multiple-choice question and the options have clear patterns in terms of the denominator and power of $$x - 1$$, we can sometimes make an educated guess by inspecting the series.In typical power series that converge, a series that has terms of the form $$(x - 1)^n$$ will converge to an interval centered at the point 1, because the series' form implies it's developed around $$x_0 = 1$$. For option (A) and option (C), the series have forms $$(x - 1)^{n+1}$$, suggesting that they are centered around $$x = 1$$ but given the additional $$n+1$$ term, they're likely to have a smaller radius of convergence.Option (B) and option (D) have the additional denominator terms of $$(n + 1)$$ and $$(n + 1)(n + 2)$$, respectively, which would make the series converge on a larger interval because the growth of the denominator will offset the growth of the numerator in the terms of the series.Since we need the series to converge at least until $$x = 2$$, we're interested in the option with the larger "cushioning" effect from the denominator to ensure that convergence holds beyond $$x = 1$$.Comparing (B) and (D), we see that the denominator of (D), which includes both $$(n + 1)$$ and $$(n + 2)$$, grows faster than that of (B), which only has $$(n + 1)(n + 2)$$ as part of its term. Thus, (D) has the added advantage in terms of a larger interval of convergence.In conclusion, without going through the detailed process of finding the interval using the ratio/root test and checking the endpoints, we can surmise that option (D) probably has the desired interval of convergence $$0 < x \leq 2$$. Nonetheless, to be completely accurate, one would need to apply the ratio test to each series and then test the endpoints to determine the exact interval of convergence for each series.
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