## Xem mẫu

30.3 Taylor Series 941 30.3 TAYLOR SERIES DeÞning Taylor Series In many examples in this chapter weve observed that the higher the degree of the Taylor polynomial generated by at , the better it approximates for near . For functions such as sin and cos , the higher the degree of the Taylor polynomial the longer the interval over which the polynomial follows the undulations of the functions graph. Letting the degree of the polynomial increase without bound gives us the Taylor series for . D e f i n i t i o n If a function has derivatives of all orders at , then the Taylor series of at (or about) is dened to be 2 , that is, . 0 EXAMPLE 30.11 SOLUTION We refer to this series as the Taylor expansion of about or centered at . In the special case where 0, the series 0 0 can be called the Maclaurin series for . From the work weve done with Taylor polynomials, we can easily nd the Maclaurin series for , sin , and cos . Find the Maclaurin series for . All derivatives of are . When evaluated at 0, is 1. Maclaurin series for : 2 1 2! ! 0 ! EXAMPLE 30.12 Find the Maclaurin series for sin . SOLUTION Even order derivatives Odd order derivatives sin sin 4sin . 21 sin 00 00 400 . 200 cos cos 5cos . 211 cos 01 01 501 . 2101 Maclaurin series for sin : 3 5 21 21 3! 5! 1 2 1! 01 2 1! 942 CHAPTER 30 Series EXAMPLE 30.13 SOLUTION Find the Maclaurin series for 1. 1 1 1 2 21 3 3 21 4 . !1 1 Maclaurin series for 1: 01 01 02 03! . 0! 1 22 3!3 4!4 ! 1 2 3 0 TheMaclaurinseriesfor 1 shouldlookfamiliar.0 isageometricserieswith 1 and % . Therefore we know that it converges to 1 for 1 and diverges for 1. This observation at the end of Example 30.13 highlights the important question What is the signicance of the Taylor series for ? For instance, for what does the Maclaurin series for sin converge? When it converges, to what does it converge? In particular, does sin 0.1 0.1 03!3 05!5 1 21! ? Or, more generally, for which values of is it true that 3 5 7 21 sin 3! 5! 7! 1 2 1! ? These latter questions can be answered using Taylors Theorem. Taking the limit as increases without bound gives lim lim . Therefore, is the sum of its Taylor series if and only if lim 0. We state this more precisely below. T h e o r e m o n C o n v e r g e n c e o f T a y l o r S e r i e s If is innitely differentiable on an interval centered around , then the Taylor series for at converges to for all if and only if lim 0 for all , where is the Taylor remainder. 30.3 Taylor Series 943 In applying this theorem we frequently use the fact that lim 0 for every . Think about this; it should make sense that eventually ! will be much larger than for xed . We prove this below. Fact: lim 0 for every real number . Proof: 0 ! Let be a positive constant integer such that 0 1. Then 1 2 3 " 1 ! " positive terms, each less than or equal to positive terms, each less than So 0 (30.4) EXAMPLE 30.14 SOLUTION If0% 1,thenlim % 0.Thereforelim % 0for0% 1andconstant. But 0 1, so lim 0. Return to (30.4) and let increase without bound. 0 lim lim 0 lim 0 0 Therefore lim 0, by the Sandwich Theorem. We are now ready to show that sin and are equal to their respective Taylor series. Show that sin 01 21! for all . For each there exists a between 0 and such that 1 1 1! Therefore 1! . The latter inequality holds because 1 sin or cos and both are bounded by 1. 1 0 lim lim 1! 0 lim 0 From the Sandwich Theorem we conclude that lim 0 and therefore lim 0 for all . Thus, sin is equal to its Taylor expansion about zero for all . 944 CHAPTER 30 Series EXAMPLE 30.15 SOLUTION Show that 0 ! for all . For each there exists a between 0 and such that 1 1 1 1! 1! is an increasing function, so . 1 0 1! 1 0 lim lim 1! But lim 1! lim 1! 0 0. 0 lim 0 So lim 0 by the Sandwich Theorem. Therefore, lim 0. We conclude that 1 2! 3! for all . EXERCISE 30.4 Show that cos is equal to its Maclaurin series for all . Take a moment to reect upon the rather remarkable results we have accumulated. Not only can we express , sin , and cos as innite polynomials (called power series), but we determined the coefcients using information about derivatives evaluated only at 0. We think of a derivative as giving local information, yet somehow information generating theentirefunctionisencodedinthesetofinnitelymanyderivatives.Thisisphilosophically intriguing. Lets take inventory on convergence issues. A Taylor series might converge to its generating function for all . For example, consider the Maclaurin series for , sin , and cos . A Taylor series might converge to its generating function only over a certain interval. For example, 1 0 only for 1, 1. At minimum a Taylor series will be equal to the value of its generating function at its center.6 Power Series Well put Taylor series in a broader context by discussing power series. 6 ItispossibleforaTaylorseriestoconverge,butnottoitsgeneratingfunction,exceptatitscenter.Thispathologyisillustrated in Problem 35 at the end of this section. 30.3 Taylor Series 945 D e f i n i t i o n A power series in is an innite series of the form 0 . A power series in ( ), or a power series centered at , is a series of the form 0 . U n i q u e n e s s T h e o r e m f o r P o w e r S e r i e s E x p a n s i o n s If has a power series expansion (or representation) at , that is, if 0 # for , then that power series is the Taylor series for at . TheUniquenessTheoremcanbeveriedbyrepeatedlydifferentiatingthepowerseries expansion term by term and evaluating each successive derivative at . The Uniqueness Theorem carries with it computational power. For example, we could have avoided computing derivatives in Example 30.13 by using the fact that 1 1 2 for 1, 1. This is a power series expansion of 1 , and therefore it must be the Taylor series for 1 at 0. Convergence of a Power Series7 T h e o r e m o n t h e C o n v e r g e n c e o f a P o w e r S e r i e s8 For a given power series 0 , one of the following is true: i. The series converges for all . ii. The series converges only when . iii. There is a number , 0 such that the series converges for all such that ( is within of the center) and diverges for all such that . is called the radius of convergence. If the series converges for all , we say ; if the series converges only at its center, we say 0. The set of all for which a power series converges is called the interval of converge of the series. From the theorem stated above we see that a power series in will have an interval of convergence centered around . At the endpoints of the interval the series could either converge or diverge; further investigation is necessary. In other words, if the radius of convergence is , the interval of convergence will be one of the following: 7 Thestudentorinstructorwhoprefersathoroughdiscussionofconvergencebeforeadiscussionoftheconvergenceofapower series can turn to page 964 (Section 30.5), and, after completing that section, return to this point. 8 Justication is given in Appendix H. ... - --nqh--