“circle of convergence”的概念、定义、翻译、参考文献-科学参考

circle of convergence

Mathematics

单词	circle of convergence
释义	circle of convergence Mathematics A circle in the complex plane such that the power series $\sum_{k = 0}^{\infty} a_{k} {(z - z_{0})}^{k}$ absolutely converges for all z satisfying \|z−z₀\|<R and diverges for all z satisfying \|z−z₀\|>R. Here R is the radius of convergence, and if the series converges for all the complex plane then R is infinite. The disc with centre z₀ and radius R is called the disc of convergence. For points on the circumference of the circle the series may either converge or diverge. These same terms apply to real power series, though the disc will in that case be an interval. The radius convergence can commonly be determined using the ratio test and in general the formula $R = \frac{1}{limsup {\| a_{k} \|}^{1 / k}}$ applies (as a consequence of Cauchy’s test).
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