Module 9 :  Infinite Series, Tests of Convergence,  Absolute and Conditional Convergence, Taylor and Maclaurin Series
Lecture 27 :  Taylor and Maclaurin series [Section 27.2]
  is its Taylor series at . Since it is a geometric series, it will be convergent if
          .

Further, its sum is
          .

Hence, has Taylor series expansion
           
(ii)
 Consider the function since
 

         ,
The Taylor (Maclaurin) Series of f at , is given by
         
By the ratio test, series converges for all , but we do not know its sum.

   
27.2.3 Theorem (Convergence of Taylor Series):
 

Let be an open interval and be a function having derivatives of all order in . For , there exists a point between and such that
       .
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