Explore BrainMass

# Cauchy- Hadamard Theorem on power series.

Not what you're looking for? Search our solutions OR ask your own Custom question.

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

Complex Variable

Cauchy- Hadamard Theorem :-
&#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.

© BrainMass Inc. brainmass.com October 6, 2022, 12:16 pm ad1c9bdddf
https://brainmass.com/math/real-analysis/cauchy-hadamard-theorem-on-power-series-37237

#### Solution Preview

Complex Variable

Cauchy- Hadamard Theorem :-
...

#### 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:
Cauchy- Hadamard Theorem :-
&#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.

\$2.49