circle of convergence

noun

1. Mathematics.,  a circle associated with a given power series such that the series converges for all values of the variable inside the circle and diverges for all values outside it.

Example Sentences

Within a common circle of convergence two power series Σ anzn, Σ bnzn can be multiplied together according to the ordinary rule, this being a consequence of a theorem for absolutely convergent series.

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In other words, the original series may legitimately be differentiated at any interior point z0 of its circle of convergence.

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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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The circle of convergence of any of the series has at least one such singular point upon its circumference.

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