The Chemistry Maths Book, Second Edition

210 Chapter 7Sequences and series Substitution into (7.19) then gives (7.20) This power series is called the MacLaurin series ex ...

7.6 MacLaurin and Taylor series 211 with general term The series (7.21) is called the binomial series, and is the generalization ...

212 Chapter 7Sequences and series (iii) For example, ifx 1 = 11 , using the first five terms of the series, ≈ 1 × 1 1.06065941 1 ...

7.6 MacLaurin and Taylor series 213 The nth derivative atx 1 = 10 isf (n) (0) 1 = 1 (−1) n− 1 (n 1 − 1 1)!, and The series conve ...

214 Chapter 7Sequences and series (7.24) This power series in(x 1 − 1 a)is called a Taylor series expansionof the functionf(x). ...

7.7 Approximate values and limits 215 If the series is terminated after a finite number of terms, the result is a polynomial app ...

216 Chapter 7Sequences and series EXAMPLE 7.14The exponential series By Taylor’s theorem (fora 1 = 10 ), where for some point 01 ...

7.7 Approximate values and limits 217 and 1.221400 1 < 1 e 0.2 1 < 1 1.221415 The exact value is 1.221403. 0 Exercise 72 L ...

218 Chapter 7Sequences and series Then becausef(a) 1 = 1 g(a) 1 = 10 , and, ifg′(a) is not zero, (7.29) This method of finding l ...

7.8 Operations with power series 219 The numerator isx 2 (1 1 + 1 x 1 + 1 x 2 221 +1-). The denominator is [(1 1 + 1 x 1 + 1 x 2 ...

220 Chapter 7Sequences and series and this can be rearranged as the Cauchy product (7.32) where The radius of convergence of the ...

7.9 Exercises 221 EXAMPLE 7.17Differentiation and integration of power series 0 Exercises 79, 80 7.9 Exercises Section 7.2 Find ...

222 Chapter 7Sequences and series Use equation (7.11) to expand in powers of x: 24.(1 1 + 1 x) 5 25.(1 1 + 1 x) 7 Calculate the ...

7.9 Exercises 223 Section 7.5 Examine the following series for convergence by Comparison test (useln 1 n 1 < 1 n): 46. 47. D’ ...

224 Chapter 7Sequences and series Section 7.7 (i) Find the MacLaurin expansion of the function(8 1 + 1 x) 123 up to terms inx ...

8 Complex numbers 8.1 Concepts We saw in Section 1.7 and in Chapter 2 that the solutions of algebraic equations are not always r ...

226 Chapter 8Complex numbers Apart from their role in extending the concept of number, complex numbers occur in several branches ...

8.2 Algebra of complex numbers 227 EXAMPLES 8.2Multiplication (i)(2 1 + 13 i)(3 1 + 14 i) 1 = 1 2(3 1 + 14 i) 1 + 13 i(3 1 + 14 ...

228 Chapter 8Complex numbers and the roots of the quadratic are the complex conjugate pair 0 Exercises 8–11 Division The divisio ...

8.3 Graphical representation 229 numbers, withy 1 = 10 , are represented by points on the xor real axisand pure imaginary number ...