radius of convergence

noun

Mathematics.

1. a positive number so related to a given power series that the power series converges for every number whose absolute value is less than this particular number.

Example Sentences

If r1 be less than the radius of convergence of a series Σ an zn and for |z| = r1, the sum of the series be in absolute value less than a real positive quantity M, it can be shown that for |z| = r1 every term is also less than M in absolute value, namely, |an| < Mr1−n.

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Starting with a convergent power series, say in powers of z, this series can be arranged in powers of z − z0, about any point z0 interior to its circle of convergence, and the new series converges certainly for |z − z0| < r − |z0|, if r be the original radius of convergence.

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If for every position of z0 this is the greatest radius of convergence of the derived series, then the original series represents a function existing only within its circle of convergence.

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