7.2 • TAYLOR SERIES REPRESENTATIONS 261For many students, it makes sense that the first series in Equations (7~14) con-verges only on the interval ( - 1, 1) because 1 !x• is undefined at the points
x = ±1. It seems unclear as to why this should also be the case for the se-ries representing 1+1;. 2 , since the real-valued function f (x) = 1 .;"' 2 is defined
everywhere. The explanation, of course, comes from the complex domain. The
complex function f (z) = l.;z• is not defined everywhere. In fact, the singular-
ities of f are at the points ±i, and the distance between them and the point
a= 0 equals 1. According to Corollary 7.3, therefore, Equations (7-12) are valid
only for z E D1 (0), and thus Equations (7-14) are valid only for x E (-1, 1). •
Alas, there is a potential fly in this ointment: Corollary 7.3 applies to Taylor
series. To form the Taylor series of a function, we must compute its derivatives.
We didn't get the series in Equations (7-12) by computing derivatives, so bow
do we know that they are indeed the Taylor series centered at a = O? Perhaps
the Taylor series would give completely different expressions from those given by
Equations (7-12). Fortunately, Theorem 7.5 removes this possibility.Thus, any power series representation off (z) is automatically the Taylor
series.
•EXAMPLE 7.5 Find the Maclaurin series off (z) = sin^3 z.
Solution Computing derivatives for f (z) would be an onerous task. Fortu-
nately, we can make use of the trigonometric identity. 3 3. 1. 3
sm z=
4
smz-
4sm z.Recall that the series for sin z (valid for all z) is sin z = f (-It (~::+ 1 ),. Using
n=O