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 Power Series
27.1.1 Definition:
(i)
 A series of the form
 

is called a power series in the variable centered at , where  for all .

(ii)
 A power series is said to converge for a particular value if
 

the series is convergent

(iii)

The set of all such that is convergent, is called the domain of convergence of the

  power series.
   
27.1.2 Examples:
(i)
 Consider the power series
 


centered at . For a fixed value of , this is a geometric series, and hence will be convergent for , with sum . Thus, we can write

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