Module 9 : Infinite Series, Tests of Convergence, Absolute and Conditional Convergence, Taylor and Maclaurin Series
Lecture 27 : Taylor and Maclaurin series [Section 27.2]
Thus, this is the Maclaurin series for . If we change the variable from to we get
the Maclaurin Series expansion for . (Note that to find Maclaurin Series
directly requires some tedious derivative computations). Now for using
theorem 27.1.8, we have .
Since for , , we have . Hence
In fact, using the alternative series test, it is easy to see that the above holds for also.
27.2.7
Algebraic operations on power series:
Suppose power series
are both absolutely convergent to and respectively Then it can be shown that the following series are absolutely convergent to
. If we change the variable from 
to 
we get 
. (Note that to find Maclaurin Series
directly requires some tedious derivative computations). Now for 
using
theorem 27.1.8, we have 
.
, 
, we have 
. Hence 
also. 
and 
respectively 
Then it can be shown that the following series are absolutely convergent to 
