Module 9 :  Infinite Series, Tests of Convergence,  Absolute and Conditional Convergence, Taylor and Maclaurin Series
Lecture 27 :  Series of functions [Section 27.1]
27.1.3 Theorem :
 

For a power series


 

precisely one of the following is true.

(i)
 The series converges only for .
(ii)
 There exists a real member such that the series converges absolutely for with , and
diverges for with
(iii)

The series converges absolutely for all .

  Proof
 

To prove the theorem, let us suppose that the series is convergent for some such that
         
Then,
        
Hence, there exists some such that
        
Then
        
Then,
        .
 
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