Radius of Convergence

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power series sum^(infty)c_kx^kwill converge only for certain values ofx例如,sum_(k=0)^(infty)x^kconverges for-1<x<1In general, there is always an interval(-R,R)in which apower seriesconverges, and the numberRis called the radius of convergence (while the interval itself is called the interval of convergence). The quantityRis called the radius of convergence because, in the case of a power series with complex coefficients, the values ofx|x|<Rform anopen diskwith radiusR

power series总是converges absolutelywithin its radius of convergence. This can be seen by fixingr=|x|and supposing that there exists asubsequence c_(n_i)such that|c_(n_i)|r^(n_i)unbounded然后power series sumc_nx^ndoes notconverge(in fact, the terms are unbounded) because it fails thelimit testTherefore, forxr=|x|>R, the power series does not converge, where


limsupdenotes thesupremum limit

Conversely, suppose thatr<RThen for any radiussr<s<R, the termsc_nx^n满足


Nlarge enough (depending ons). It is sufficient to fix a value forsin betweenrRBecauses/R<1, the power series is dominated by a convergentgeometric series. Hence, thepower seriesconverges absolutely by thelimit comparison test

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