Maclaurin and Taylor Series

Transcription

1 Maclaurin and Talor Series 6.5 Introduction In this block we eamine how functions ma be epressed in terms of power series. This is an etremel useful wa of epressing a function since (as we shall see) we can then replace complicated functions in terms of simple polnomials. The onl requirement (of an significance) is that the complicated function should be smooth; this means that at a point of interest, it must be possible to differentiate the function as often as we please. Prerequisites Before starting this Block ou should... Learning Outcomes After completing this Block ou should be able to... find the Maclaurin and Talor series epansions of given functions find Maclaurin epansions of functions b combining known power series together find Maclaurin epansions b using differentiation and integration have knowledge of power series and of the ratio test 2 be able to differentiate simple functions 3 be familiar with the rules for combining power series Learning Stle To achieve what is epected of ou... allocate sufficient stud time briefl revise the prerequisite material attempt ever guided eercise and most of the other eercises

2 . Maclaurin and Talor Series As we shall see, man functions can be represented b power series. In fact we have alread seen in earlier Blocks eamples of such a representation. For eample, = < ln( + ) = < e = ! ! all The first two eamples show that, as long as we constrain to lie within the domain < (or, equivalentl, <<), then in the first case has the same numerical value as and in the second case ln(+) has the same numerical value as In the third eample we see that e has the same numerical value as but in 2! this case there is no restriction to be placed on the value of since this power series converges for all values of. The following diagram shows this situation geometricall. As more and more terms are used from the series the curve representing 2! 3! e is better and better approimated. In (a) we show the linear approimation to e. In (b) and (c) we show, respectivel, the quadratic and cubic approimations ! ! + 3 3! e e e (a) (b) (c) These power series representations are etremel important, from man points of view. Numericall, we can simpl replace the function b the quadratic epression as long as is so small so that powers of greater than or equal to 3 can be ignored in comparison to quadratic terms. This approach can be used to approimate more complicated functions in terms of simpler polnomials. Our aim now is to see how these power series epansions are obtained. 2. The Maclaurin Series Consider a function f() which can be differentiated at = 0 as often as we please. For eample e, cos, sin would fit into this categor but would not. Let us assume that f() can be represented b a power series in : f() =a 0 + a + a a a = a p p p=0 Engineering Mathematics: Open Learning Unit Level 2

3 where a 0,a,a 2,... are constants to be determined. If we substitute = 0 then, clearl f(0) = a 0 The other constants can be determined b further differentiating and, on each differentiation, substituting = 0. For eample, differentiating once: so, putting =0,wehavef (0) = a. Continuing to differentiate: f () =0+a +2a 2 +3a a f () =0+2a 2 + 3(2)a 3 + 4(3)a so Further: so f (0)=2a 2 or a 2 = 2 f (0) f () = 3(2)a 3 + 4(3)(2)a f (0) = 3(2)a 3 impling a 3 = 3(2) f (0) Continuing in this wa we easil find that (remembering that 0! = ) a n = n! f (n) (0) n =0,, 2,... where f (n) (0) means the value of the n th derivative at = 0 and f (0) (0) means f(0). Bringing all these results together we find Ke Point If f() can be differentiated as often as we please: f() =f(0) + f (0) + 2 2! f (0) + 3 3! f (0) +...= This is called the Maclaurin epansion of f() p=0 p! f (p) (0) p Eample Find the Maclaurin epansion of cos. 3 Engineering Mathematics: Open Learning Unit Level

4 Solution Here f() = cos and, differentiating a number of times: f() = cos, f () = sin, f () = cos, f () = sin etc. Thus, evaluating each of these at =0: f(0)=, f (0)=0, f (0) =, f (0) = 0 etc. Now, substituting into f() =f(0) + f (0) + 2 2! f (0) + 3 3! f (0) +..., implies cos = 2 2! + 4 4! 6 6! +... The reader should confirm (b finding the radius of convergence) that this series is convergent for all values of. The geometrical approimation to cos b the first few terms of its Maclaurin series are shown in the following diagram. 2 2! + 4 4! cos cos 2 2! cos Tr each part of this eercise Find the Maclaurin epansion of ln( + ). (Note that we cannot find a Maclaurin epansion of the function ln since this function cannot be differentiated at = 0). Part (a) Find the first few derivatives of f() = ln( + ) Part (b) Now obtain f(0), f (0), f (0), f (0),... Part (c) Hence, obtain the Maclaurin epansion of ln( + ). Part (d) Now obtain the interval of convergence? Note that when = ln2= so the alternating harmonic series converges to ln , a claim first made in Block 2. Engineering Mathematics: Open Learning Unit Level 4

5 Eample Find the Maclaurin epansion of e ln( + ). Solution Here, instead of finding the derivatives of f() =e ln( + ), we can multipl together the Maclaurin epansions we alread know: e = ! all 3! and ln( + ) = < 3 The resulting power series will onl be convergent if <. That is ) e ln( + ) = (+ + )( 2 2! + 3 3! = = < 40 (You must take care not to miss relevant terms when carring through the multiplication). The Maclaurin epansion of a product of two functions: f()g() is obtained b multipling together the Maclaurin epansions of f() and of g() and collecting like terms together. The product series will have a radius of convergence equal to the smaller of the two separate radii of convergence. Tr each part of this eercise Find the Maclaurin epansion of cos 2 up to powers of 4. Hence write down the epansion of sin 2 to powers of 6. Part (a) First, write down the epansion of cos Part (b) Now, b multiplication, find the epansion of cos 2. (The reader could tr to obtain the power series epansion for cos 2 b using the trigonometric identit cos 2 = ( + cos 2)). 2 Part (c) Now obtain the epansion of sin 2. 5 Engineering Mathematics: Open Learning Unit Level

6 3. Differentiation of Maclaurin Series We have alread noted that, b the binomial series, = < Thus, with replaced b + = < Also, we have obtained the Maclaurin epansion of ln( + ): ln( + ) = < 4 Now, we differentiate both sides with respect to : + = This demonstrates that the Maclaurin epansion of a function f() ma be differentiated term b term to give a series which will be the Maclaurin epansion of df. d As we noted in block 4 the derived series will have the same radius of convergence as the original series. Tr each part of this eercise Find the Maclaurin epansion of ( ) 3. Part (a) First write down the epansion of ( ) Part (b) Now, b differentiation, obtain the epansion of ( ) 2 Part (c) Differentiate again to obtain the epansion of ( ) 3. The final series: has radius of convergence R = since the original series, before differentiation, has this radius of convergence (but this can also be found directl using the formula R = lim a n n a n+ and using the fact that the coefficient of the nth term is a n = n(n +)). 2 Engineering Mathematics: Open Learning Unit Level 6

7 4. The Talor Series The Talor series is a generalisation of the Maclaurin series being a power series developed in powers of ( 0 ) rather than in powers of. Thus Ke Point If the function f() can be differentiated as often as we please at = 0 then: f() =f( 0 )+( 0 )f ( 0 )+ ( 0) 2 f ( 0 )+... 2! This is called the Talor series of f() about the point 0. The reader will see that the Maclaurin epansion is obtained if 0 is chosen to be zero. Tr each part of this eercise Obtain the Talor series epansion of of ( 2)). about = 2. (That is, find a power series in powers Part (a) First, obtain the derivatives of f() = Part (b) Now evaluate these derivatives at 0 =2. Part (c) Hence, write down the Talor epansion of f() = about =2 The reader should confirm that this series is convergent if 2 <. In the diagram following some of the terms from the Talor series are plotted to compare with ( ). +( 2) ( 2) 2 +( 2) ( 2) +( 2) ( 2) 2 7 Engineering Mathematics: Open Learning Unit Level

8 5. Computer Eercise or Activit For this eercise it will be necessar for ou to access the computer package DERIVE. DERIVE can be used to obtain the Maclaurin series epansion of most functions. For eample to obtain the Maclaurin epansion of we would ke Author: Epression /( ). DERIVE ( ) responds with. Now ke Calculus:Talor series. In the bo presented choose as Variable, then 0 as the Epansion Point and (sa) 5 as Order. Then on hitting the Simplif button DERIVE responds as epected. To obtain a Talor series (i.e. epansion about some point other than 0) is a straightforward eercise; just choose the appropriate value for the Epansion Point. However, note that DERIVE alwas presents the Talor series in powers of (which is not the wa we have presented the epansion in the tet). You need to be a little careful here. If ou want the Talor epansion of (sa) about the point = 2 to order (sa) 5 then DERIVE will obtain the epression (as we have in the tet) = +( 2) ( 2)2 +( 2) 3 ( 2) 4 +( 2) 5 and then epand this to present it as: The coefficients of the powers of will necessaril change as the order changes. For eample to epand the same function about the same point ( = 2) to order 6 will produce the response from DERIVE: which is the epanded form of +( 2) ( 2) 2 +( 2) 3 ( 2) 4 +( 2) 5 ( 2) 6 Engineering Mathematics: Open Learning Unit Level 8

9 End of Block Engineering Mathematics: Open Learning Unit Level

10 f () = +, f () = ( + ) 2, f () = 2 ( + ), 3 f(n) () = ( )n+ (n )! ( + ) n Engineering Mathematics: Open Learning Unit Level 0

11 f(0)=0 f (0)=, f (0) =, f (0)=2, f (n) (0)=( ) n+ (n )! Engineering Mathematics: Open Learning Unit Level

12 ln( + ) = ( )n+,...+ n +... This has alread been obtained in Block n Engineering Mathematics: Open Learning Unit Level 2

13 R =. Also at = the series is convergent (alternating harmonic series) and at = the series is divergent. Hence this Maclaurin eapnsion is onl valid if <. The geometrical closeness of the polnomial terms with the function ln( + ) for < are displaed in the following diagram ln( + ) ln( + ) ln( + ) Engineering Mathematics: Open Learning Unit Level

14 cos = 2 2! + 4 4! +... Engineering Mathematics: Open Learning Unit Level 4

15 ( )( ) cos 2 = ! 4! 2! 4! =( )+( + 4 2! 4! 2! 4...)+(4...)+ = 4! Engineering Mathematics: Open Learning Unit Level

16 ( ) sin 2 = cos 2 = = Engineering Mathematics: Open Learning Unit Level 6

17 = < 7 Engineering Mathematics: Open Learning Unit Level

18 ( ) 2 = d d ( )= Engineering Mathematics: Open Learning Unit Level 8

19 = d ( ) 3 2 d ( ) = [2 + 6 ( ) ]= Engineering Mathematics: Open Learning Unit Level

20 f () = ( ) 2, f () = 2 ( ) 3,... f (n) () = n! ( ) n+ Engineering Mathematics: Open Learning Unit Level 20

21 f (2)=, f (2) = 2,... f (n) (2)=( ) n+ n! 2 Engineering Mathematics: Open Learning Unit Level

22 = +( 2) ( 2)2 +( 2) Engineering Mathematics: Open Learning Unit Level 22