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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--