# Cauchy- Hadamard Theorem on power series.

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Complex Variable

Cauchy- Hadamard Theorem :-

∞

For every power series ∑ anzn there exist a number R, 0 ≤ R < ∞

n=0

called the radius of convergence with the following properties:

(i) The series converges absolutely for every |z| < R

(ii) If 0 ≤ ρ < R, the series converges uniformly for |z| ≤ ρ

(iii) If |z| > R, the terms of the series are unbounded and the series is consequently divergent.

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##### Solution Summary

∞

This solution is comprised of a detailed explanation of the radius of convergence of the power series ∑ anzn.

n=0

It contains step-by-step explanation for the following problem:

Cauchy- Hadamard Theorem :-

∞

For every power series ∑ anzn there exist a number R, 0 ≤ R < ∞

n=0

called the radius of convergence with the following properties:

(i) The series converges absolutely for every |z| < R

(ii) If 0 ≤ ρ < R, the series converges uniformly for |z| ≤ ρ

(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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Complex Variable

Cauchy- Hadamard Theorem :-

...

###### Education

- BSc, Manipur University
- MSc, Kanpur University

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