radius of convergence

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.

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.

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.