2024 JPJC Chapter 11 Maclaurin Series and Binomial Expansion

Jurong Pioneer Junior College H2 Mathematics (9758) JC2-2024 Chapter 11 Maclaurin Series and Binomial Expansion (Students’ version) / Pg 1 Chapter 11: Maclaurin’s Series & Binomial Expansions Content Outline  standard series expansion of (1 )nx for any rational n, ex, sinx, cosx and ln(1 )x  derivation of the first few terms of the Maclaurin’s series by – repeated differentiation, e.g. secx – repeated implicit differentiation, e.g. 3 2 22y y y x x    – using standard series, e.g. e cos 2xx, 1ln1xx     range of values of x for which a standard series converges  concept of “approximation”  small angle approximations: sinx x, 21cos 12x x , tanx x Exclude derivation of the general term of the series. References  Websites https://www.h2maths.site/  Books (1) Ho Soo Thong, Tay Yong Chiang & Koh Khee Meng, “College Mathematics Syllabus C Volume 1”, Pan Pacific Publications, Call Number: HO510 (2) Pure Mathematics by Alan Sherlock, Elizabeth Roebuck, Timothy Heneage, Shirley Beck: Chapter 16 (3) Pure Mathematics 2 by L Bostock, S Chandler: Chapter 5 Pre-requisites: (1) Differentiation techniques including implicit differentiation (Chapter 6) (2) O level knowledge: Binomial Theorem, Sine Rule, Cosine Rule, Inequalities

Jurong Pioneer Junior College H2 Mathematics (9758) JC2-2024 Chapter 11 Maclaurin Series and Binomial Expansion (Students’ version) / Pg 2 1. Introduction A power series in x is a series of ascending powers of x of the form 2 3 0 1 2 3f ( ) ... ... n nx a a x a x a x a x       where ...,,...,,,, 3210 naaaaa are constants. Note that a polynomial is a subset of a power series, as a polynomial has only finite number of terms with positive integer powers. Examples of Power Series: 2 4x , 25 4x x  , 33 5x x  , 4 3 21 5 102x x x x    , 1 2 3(1 ) 1 .... , 1x x x x x       Example of Non-power Series: 3 2 24 7x xx  Reason: The second term contains a _____________________ of x, and the third term contains a ______________________ for x. In this chapter, we shall learn to approximate a function by using polynomials. 2. Maclaurin’s Series Key Result: ( ) 2 3f (0) f (0) f (0)f ( ) f (0) f (0) ... ...2! 3! ! n n'' '''x x x x x n       Note that ! 1 2 3 ... ( 1)r r r       Remarks: (1) The above formula is given in the formula list MF26 as 2 ( )f ( ) f (0) f (0) f (0) ... f (0) ...2! ! n nx xx x '' n      (2) The proof is in the Appendix. Example 1 Find the Maclaurin’s expansion for f(x) =ex up to and including the term in4x . Solution: f(x) = ex f '( )x = f ''( )x = f '''( )x = f (4)(x) = f(0) = f '(0)= f ''(0)= f '''(0)= f(4)(0) = Therefore, f ( )x  2 3 4 1 2 6 24 x x xx    

Jurong Pioneer Junior College H2 Mathematics (9758) JC2-2024 Chapter 11 Maclaurin Series and Binomial Expansion (Students’ version) / Pg 3 Remark: The Maclaurin’s series may be used for approximations. In general, an approximation becomes better when more terms of the Maclaurin’s series of f ( )x are included as illustrated below. Let ( )np x denote a polynomial of up to degree n. By successively differentiating ex , we find the Maclaurin’s series expansion of ex as a polynomial of (i) degree 0, 0( )p x , (ii) degree 1, 1( )p x , (iii) degree 2, 2( )p x , (iv) degree 3, 3( )p x . Let f ( ) exx  f (0) 1  f ( ) exx  f '(0) 1  f ( ) exx  f ''(0) 1  f ( ) exx  f '''(0) 1  From MF26, 2 3 f ( ) f (0) f '(0) f "(0) f (0)2! 3! x xx x        2 3 2 3 e 1 (1) (1) (1)2! 3! 1 2 6 x x xx x xx             (i) 0e ( )x p x  (ii) 1e ( )x p x  (iii) 2e ( )x p x  (iv) 3e ( )x p x  Question: Which of the above polynomials would give us the best approximation for ex ?

Jurong Pioneer Junior College H2 Mathematics (9758) JC2-2024 Chapter 11 Maclaurin Series and Binomial Expansion (Students’ version) / Pg 4 We now use graphs to compare the accuracy of the approximations of ex made by these four polynomials. Compare the graph of exy and the graphs of the four polynomials. Note that the graphs of 1( )y p x, 2( )y p x and 3( )y p x are virtually indistinguishable from the graph of exy near 0x. So these polynomials are good approximations of ex for values of x which are close to 0. However, the farther x is from 0, the poorer these approximations become. For a larger interval of x from 0, the approximation made by ( )np x gets better as n gets larger. In other words, the approximation becomes better when the polynomial includes more terms of the Maclaurin’s series of ex. Thus, ________ give us the best approximation for ex. y x 0 1 2 1 5 4 3 2

Jurong Pioneer Junior College H2 Mathematics (9758) JC2-2024 Chapter 11 Maclaurin Series and Binomial Expansion (Students’ version) / Pg 5 Example 2 Find the first 3 non-zero terms in the Maclaurin’s expansion for (i) f( x) = sin x (ii) y = ln (1 + x) Solution: (i) f( x) = sin x f(0) = f '( )x = f '(0)= f ''( )x = f ''(0)= f '''( )x = f '''(0)= f (4)(x) = f (4)(0) = f (5)(x) = f (5)(0) = Therefore, f ( )x  2 3 4 5 0 (1) (0) ( 1) (0) (1)2! 3! 4! 5! x x x xx        3 5 6 120 x xx   (ii) y = f(x) = ln (1 + x) f(0) = f '( )x = f '(0)= f ''( )x = f ''(0)= f '''( )x = f '''(0)= Therefore, y = 2 3 0 (1) ( 1) (2)2! 3! x xx      2 3 2 3 x xx  

Jurong Pioneer Junior College H2 Mathematics (9758) JC2-2024 Chapter 11 Maclaurin Series and Binomial Expansion (Students’ version) / Pg 6 Example 3 Given that cos ln(1 )y x  , prove that (i)  d(1 ) sin ln(1 ) ,d yx xx   (ii) 2 2 2 d d(1 ) (1 ) 0d d y yx x yx x     . Obtain an equation relating 3 2 3 2 d d d, and d d d y y y x x x . Hence find Maclaurin’s series for y, up to and including the term in 3x . Solution (i) Let cos ln(1 )y x  Then d d y x=   d1 sin ln(1 )d yx xx   (shown) (ii) Differentiate (i) w.r.t. x:    22 2 d d1 1 0d d y yx x yx x     (shown) Note: y =cos ln(1 )x . Differentiate (ii) wrt x :    3 22 3 2 d d d1 3 1 2 0d d d y y yx xx x x     When x = 0, y = 1, d d y x0, 2 2 d d y x  1 and 3 3 d d y x  3 By Maclaurin’s Series, 2 3 31 ...2! 3! x xy    2 3 1 2 2 x x  

Jurong Pioneer Junior College H2 Mathematics (9758) JC2-2024 Chapter 11 Maclaurin Series and Binomial Expansion (Students’ version) / Pg 7 3. Standard Series Expansions The following are standard series expansions derived from the Maclaurin’s series (discussed in Section 2) and the ranges of values of x for which the expansions are valid. They are found in MF26 and may be quoted without proof unless their derivation is asked for. 2( 1) ( 1)...( 1)(1 ) 1 2! ! n r n n n n n rx nx x x r             (| | < 1)x 2 3 e 1 2! 3! ! r x x x xx r        (all x) 3 5 2 1 ( 1)sin 3! 5! (2 1)! r rx x xx x r         (all x) 2 4 2 ( 1)cos 1 2! 4! (2 )! r rx x xx r        (all x) 2 3 1 ( 1)ln(1 ) 2 3 r rx x xx x r         ( 11  x ) Note that when we deal with the sum or product of standard Maclaurin’s series, the range of values of x for which the expansion is valid is the intersection of the ranges of values of x for which each standard Maclaurin’s series used is valid. Example 4 By using the standard series, expand e sinxy x as a series of ascending powers of x up to and including the term in x3. Solution e sinxy x 3