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In mathematics, the radius of convergence of a power series is the radius of the largest disk in which the series converges. It is either a non-negative real number or ∞. When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal to the radius of convergence, and it is the Taylor series of the analytic function to which it converges. ==Definition== For a power series ''ƒ'' defined as: : where :''a'' is a complex constant, the center of the disk of convergence, :''c''''n'' is the ''n''th complex coefficient, and :''z'' is a complex variable. The radius of convergence ''r'' is a nonnegative real number or ∞ such that the series converges if : and diverges if : Following this definition we get another representation: In other words, the series converges if ''z'' is close enough to the center and diverges if it is too far away. The radius of convergence specifies how close is close enough. On the boundary, that is, where |''z'' − ''a''| = ''r'', the behavior of the power series may be complicated, and the series may converge for some values of ''z'' and diverge for others. The radius of convergence is infinite if the series converges for all complex numbers ''z''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「radius of convergence」の詳細全文を読む スポンサード リンク
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