circle of convergence

American  

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.

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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.

From Project Gutenberg

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

From Project Gutenberg

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.

From Project Gutenberg

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.

From Project Gutenberg

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

From Project Gutenberg