Module 9 : Infinite Series, Tests of Convergence, Absolute and Conditional Convergence, Taylor and Maclaurin Series
Lecture 27 : Taylor and Maclaurin series [Section 27.2]
Similarly, one can show .
27.2.5
Note:
(i)
Suppose, a power series
is convergent is an open interval I around a point a , and ----------(*)
A natural question arises, in the power series above the Taylor series expansion of The answer is yes. In fact if (*) holds, then the power series has nonzero radius of
convergence and hence by theorem 27.1.7, series can be differentiated term by term,
giving
....
Thus
(ii)
In view of (i) above, if is expressed as sum of a power series, then it must be Taylor series of .
Thus, technique of previous section can be used to find Taylor
series expansions.
.
, and 
----------(*)
The answer is yes. In fact if (*) holds, then the power series has nonzero radius of
convergence and hence by theorem 27.1.7, series can be differentiated term by term,
giving 
....
