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It is known that the power series $ \sum_{n=0}^\infty a_n x^n$ and $ \sum_{n=0}^\infty n a_n x^{n-1}$ have the same radius of convergence $ r$ . Is it true that if $ r<\infty$ and $ \sum_{n=0}^\infty na_n r^{n-1}$ converges then $ \sum_{n=0}^\infty a_n r^n$ converges, too? The same question for $ -r$ . If not,Read more