Chapter 8: Infinite Sequences and Series
Section 8.5: Taylor Series
Taylor and Maclaurin Series
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 fx, 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)
which is formed by the "Taylor series recipe" may not converge to fx. (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 fz, 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
(not just Rnx=fn+1cn+1!x−an+1 from Theorem 3.3.1) goes to zero as n→∞, then the formal Taylor series of fx does indeed converge to, i.e., represent, fx. But it is no easy matter to show this for an arbitrary function, essentially because of the need to have either a representation of fnx or an estimate of how these derivatives behave as n→∞. 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 Binomial Series
The function fx=1+xc has a Maclaurin series for any real number c and x<1. The series is a generalization of the Binomial expansion where c=k, for positive integers k.
For c=k, a positive integer, the expansion is of the form 1+xk=∑n=0knkxn, where nk=n!k! n−k!.
For real c, the expansion is then 1+xc=∑n=0∞cnxn, where now cn=c⋅c−1⋅⋯⋅c−n−1n!, and c0=1.
Table 8.5.1 lists the first few values of cn.
Table 8.5.1 First few values of cn
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.
Show that the formal Taylor expansion of fx=e−1/x2x≠00x=0is identically zero, so that this expansion does not represent fx. Hint: Show that fn0=0 for n=1,2,….
In Examples 8.5.2-7, show that R^n+1x for the given function fx goes to zero as n→∞, establishing that f has a Maclaurin series. Find the terms of that series.
For each function in Examples 8.5.8-10, use the Binomial expansion formula to obtain its Maclaurin series.
Expand the integrand in ∫01sinx2 ⅆx and integrate termwise to obtain an estimate of the integral guaranteed correct to three decimal places. Compare to the value Maple provides.
Expand the integrand in ∫01J0x ⅆx and integrate termwise to obtain an estimate of the integral guaranteed correct to three decimal places. Compare to the value Maple provides.
Obtain the Maclaurin series for e3 x2 by manipulating the series for ex.
Obtain the Maclaurin series for x2cos2 x by manipulating the series for cosx.
Obtain the Maclaurin series for x/1−x2 from an appropriate geometric series.
Obtain the Maclaurin series for 1−x21+x2 from appropriate geometric series.
Obtain p1x and p2x, degree 8 Maclaurin polynomials for fx=ex and gx=ln1+x, respectively.
Form the product p1⋅p2.
Obtain the degree 11 Maclaurin polynomial for the product fx⋅gx.
Obtain the first 10 coefficients cn formed from the Cauchy product of the coefficients of p1 and p2.
Observe that the coefficients formed from the Cauchy product (Theorem 8.2.4) are always correct, but those formed from the product of the polynomials are correct only to a limited degree in x.
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