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High Quality Content by WIKIPEDIA articles! Every power series with a positive radius of convergence is analytic on the interior of its region of convergence. All holomorphic functions are complex-analytic. Sums and products of analytic functions are analytic, as are quotients as long as the denominator is non-zero. If a power series with radius of convergence r is given, one can consider analytic continuations of the series, i.e. analytic functions f which are defined on larger sets than { x : x - c r } and agree with the given power series on this set. The number r is maximal in the following sense: there always exists a complex number x with x - c = r such that no analytic continuation of the series can be defined at x.