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

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 .

 
27.1.4 Definition :
 

The radius of convergence of a power series   is defined to be number

(i)
 if the series is divergent for all .
(ii)
 if, the series is absolutely convergent for all .
(iii)
 , the positive member such that the series diverges a for all such that and the series converges
 

absolutely for all such that . The interval such that the series converges is convergent for all is called the interval of convergence .

27.1.5 Remark :
 

Note that the interval of convergence is either a singleton set, or a finite interval or the whole real line. In case it is a finite interval, the series may or may not converge at the and points of this interval. At all interior points of this interval, the series is absolutely convergent.

27.1.6

Example :

  Consider the power series
1
5