Theorem 3.3.1 is a statement of Taylor's theorem, expressing a sufficiently smooth function as the sum of a polynomial and a remainder term.

Functions for which the remainder term goes to zero for all $x$ in some interval about the expansion point are essentially given by an "infinite polynomial" or, in terms of Chapter 8, by an infinite series. Thus, a function with an appropriately behaved remainder has a power series representation, and this series is called a Taylor series.

When the expansion point is $x\=0$, the power series representation of $f$ is sometimes called a Maclaurin series, but some authors will simplify the terminology and use just the term "Taylor series" for all convergent power-series.

Thus, if a power series converges to $f\left(x\right)$, then that series is the Taylor series of $f$. But given an arbitrary function $f$, even one for which all derivatives exist, the expansion (called the formal Taylor expansion)

$\sum _{n\=0}^{\infty }\frac{f^{\left(n\right)}\left(c\right)}{n\!}{\left(x-c\right)}^n$ 

which is formed by the "Taylor series recipe" may not converge to $f\left(x\right)$. (See Example 8.5.1.)

Once again, a convergent power series is the Taylor series for the limit function, but the formal Taylor expansion of $f$ may not be the power-series representation of $f$. because $f$ may not have a power-series representation.

Section 8.5 deals with functions that indeed have a Taylor series representation. Determining which functions actually have a power-series (and hence a Taylor series) representation is no small matter. The most satisfying answers to this question are given for functions of a complex variable, that is, for functions $f\left(z\right)$, where $z\=x\+i y$. For such functions, if one derivative exists in a neighborhood, all derivatives exist and the Taylor expansion actually represents the function. But for functions of the real variable $x$, the situation is not so sanguine. Real functions can have just a finite number of derivatives and no more. Moreover, even functions with all derivatives may not have a Taylor series that converges back to the function, as is the case with the example function in Example 8.5.1.

If it can be shown that the Taylor-expansion remainder

 ${\stackrel{\^}{R}}_n\left(x\right)\=\frac{f^{\left(n\+1\right)}\left(x\right)}{\left(n\+1\right)\!}{\left(x-a\right)}^{n\+1}$ 

(not just  $R_n\left(x\right)\=\frac{f^{\left(n\+1\right)}\left(c\right)}{\left(n\+1\right)\!}{\left(x-a\right)}^{n\+1}$ from Theorem 3.3.1) goes to zero as $n\to \infty$, then the formal Taylor series of $f\left(x\right)$ does indeed converge to, i.e., represent, $f\left(x\right)$. But it is no easy matter to show this for an arbitrary function, essentially because of the need to have either a representation of $f^{\left(n\right)}\left(x\right)$ or an estimate of how these derivatives behave as $n\to \infty$. In the examples below, this is done for a few of the elementary functions, but in general, determining whether or not a function has a Taylor series representation relies heavily on the theory of complex variables.

The function $f\left(x\right)\&equals;{\left(1\&plus;x\right)}^c$ has a Maclaurin series for any real number $c$ and $\left|x\right|<1$. The series is a generalization of the Binomial expansion where $c\&equals;k$, for positive integers $k$.

For $c\&equals;k$, a positive integer, the expansion is of the form ${\left(1\&plus;x\right)}^k\&equals;\sum _{n\&equals;0}^k\left(\genfrac{}{}{0ex}{}{n}{k}\right)x^n$, where $\left(\genfrac{}{}{0ex}{}{n}{k}\right)\&equals;\frac{n\&excl;}{k\&excl; \left(n-k\right)\&excl;}$.

For real $c$, the expansion is then ${\left(1\&plus;x\right)}^c\&equals;\sum _{n\&equals;0}^{\infty }\left(\genfrac{}{}{0ex}{}{c}{n}\right)x^n$, where now $\left(\genfrac{}{}{0ex}{}{c}{n}\right)\&equals;\frac{c\cdot \left(c-1\right)\cdot \cdots \cdot \left(c-\left(n-1\right)\right)}{n\&excl;}$, and $\left(\genfrac{}{}{0ex}{}{c}{0}\right)\&equals;1$.

Table 8.5.1 lists the first few values of $\left(\genfrac{}{}{0ex}{}{c}{n}\right)$.

The binomial coefficient in Maple is generalized to include the case where $c$ is not an integer. A template for this function is available in the Expression palette, and this template is equivalent to invoking Maple's binomial command.