Module 9 : Infinite Series, Tests of Convergence, Absolute and Conditional Convergence, Taylor and Maclaurin Series
Lecture 27 : Taylor and Maclaurin series [Section 27.2]
27.2
Taylor Series and Maclaurin series
In section we saw that a function can be approximated by a polynomial of degree depending upon its order of smoothness. If the error terms converge to zero, we set a special power series expansion for .
27.2.1
Definition:
Let be a function which has derivative of all order in . Then the power series
is called the Taylor series for at . We say has Taylor series expansion at , if its Taylor series is convergent for and its sum is . For , the Taylor series for is called the Maaculurin Series for at .
27.2.2
Examples:
(i)
For the function its derivatives of all order exist in domain
depending upon its order of smoothness. If the error terms converge to zero, we set a special power series expansion for 
. 
be a function which has derivative of all order in 
. Then the power series 
. We say 
and its sum is 
. For 
, the Taylor series for 
. 
its derivatives of all order exist in domain
. 
,
.
