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)