Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. ?(x/23)^k The radius of convergence is R =
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First, we need to use the ratio test to find the radius of convergence: $$\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=\lim_{n\to\infty}\left|\frac{2^{n+1}(x-1)^{n+1}}{(n+1)!}\cdot\frac{n!}{2^n(x-1)^n}\right|=\lim_{n\to\infty}\frac{2|x-1|}{n+1}=0$$ Since the Show more…
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