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# Cauchy- Hadamard Theorem on power series.

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Complex Variable

&#8734;
For every power series &#8721; anzn there exist a number R, 0 &#8804; R < &#8734;
n=0

called the radius of convergence with the following properties:

(i) The series converges absolutely for every |z| < R
(ii) If 0 &#8804; &#961; < R, the series converges uniformly for |z| &#8804; &#961;
(iii) If |z| > R, the terms of the series are unbounded and the series is consequently divergent.

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Complex Variable

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#### Solution Summary

&#8734;
This solution is comprised of a detailed explanation of the radius of convergence of the power series &#8721; anzn.
n=0
It contains step-by-step explanation for the following problem:
&#8734;
For every power series &#8721; anzn there exist a number R, 0 &#8804; R < &#8734;
n=0

called the radius of convergence with the following properties:

(i) The series converges absolutely for every |z| < R
(ii) If 0 &#8804; &#961; < R, the series converges uniformly for |z| &#8804; &#961;
(iii) If |z| > R, the terms of the series are unbounded and the series is consequently divergent.

Solution contains detailed step-by-step explanation.

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