This series is unique; it is called the Taylor series of at . The cofficients of this series are determined by the following formula:
The Taylor series of a function about 0 is called the Maclaurin series of .
We can use Taylor series in order to find limits:
Using 1.2, we get:
The following result is a consequence of Thm 3.5 and Thm 3.7.
Important remark: When we studied power series over the reals we had a surprise: the convergence domain of a power series is not always obvious.
Take . This function is defined over . Its first MacLaurin expansions are given by:
It seems that the visualization shows that the successive approximations tend to the original function only for . The condition for a geometric sequence to be convergent supports this impression. First of all, the function is not defined at 1 (where it has a singular point) and this point acts as a "barrier". But, does the power series make sense out of the interval ? Actually not in our frame of study. Maybe in other frames.
Now take . It is obtained by the substitution of instead of . The first MacLaurin expansions are given by:
We have here the same visual impression: the MacLaurin series tends to the given function for . But for the function , -1 and 1 are not points of discontinuity. So, what happens?
Passing to the complex setting, consider the function of the complex variable given by . It is defined over . The corresponding MacLaurin series is given by and is convergent for in the open unit ball centered at the origin, i.e. on the largest ball centered at 0 at not touching the two points where fails to be defined (see Figure 3).
This example shows the importance of working in a complex setting. Without exaggeration, we could say that the complex setting is "more natural" than the real one.
Noah Dana-Picard 2007-12-24