\begin{tabular}{|l|l|l|l|} \hline Entered & Answer Preview & Result & Message \\ \hline\( (-0.25,0.25] \) & \( \left(\frac{-1}{4}, \frac{1}{4}\right] \) & incorrect & The type of interval is incorrect \\ \hline \end{tabular} The answer above is NOT correct. Consider the series \[ \sum_{n=1}^{\infty} \frac{(4 x)^{n}}{n} \] Find the interval of convergence of this power series by first using the ratio test to find its radius of convergence and then testing the series' behavior at the endpoints of the interval specified by the radius of convergence. interval of convergence \( =(-1 / 4,1 / 4] \)
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The ratio test states that a series \(\sum a_n\) converges if the limit \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\) is less than 1. For our series, \(a_n = \frac{(4x)^n}{n}\), so \(a_{n+1} = \frac{(4x)^{n+1}}{n+1}\). Show more…
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