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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