The Chemistry Maths Book, Second Edition

190 Chapter 6Methods of integration 72.If , show that (Equation (6.33)) Evaluate by means of the substitut ...

7 Sequences and series 7.1 Concepts A series is a set of terms that is to be summed. The terms can be numbers, variables, functi ...

192 Chapter 7Sequences and series form a sequence defined by the general term u r 1 = 111 + 1 2(r 1 − 1 1), r 1 = 1 1, 2, 3,1= A ...

7.2 Sequences 193 (iii) Harmonic sequence: (iv) Fibonacci sequence (‘series’): 2 1, 1, 2, 3, 5, 8, 13,= u r+ 2 1 = 1 u r+ 1 1 + ...

194 Chapter 7Sequences and series limit is not finite or not unique the sequence is divergent. For example, the arithmetic progr ...

7.2 Sequences 195 EXAMPLES 7.2Limits. (i) We have so that. (ii) Dividing top and bottom byr 2 , (iii) (iv) The Fibonacci sequenc ...

196 Chapter 7Sequences and series 7.3 Finite series Given a sequenceu 1 , u 2 , u 3 , =the partial sums S 1 1 = 1 u 1 S 2 1 = 1 ...

7.3 Finite series 197 Then, addition of the two forms, term by term, gives 2 S n 1 = 1 [2a 1 + 1 (n 1 − 1 1)d] 1 + 1 [2a 1 + 1 ( ...

198 Chapter 7Sequences and series This equation can be regarded in two ways: (i) the value of the sum(1 1 + 1 x 1 + 1 x 2 1 +1-1 ...

7.3 Finite series 199 The coefficient ofx r in the expansion (7.11) is (7.12) and is called a binomial coefficient(sometimes rea ...

200 Chapter 7Sequences and series (ii) By equation (7.14), 0 Exercises 26–33 The binomial coefficients form a pattern of numbers ...

7.3 Finite series 201 are important in combinatorial theory (Section 21.6) and are used in a popular deriva- tion of the Boltzma ...

202 Chapter 7Sequences and series 0 Exercises 36–39 Some finite series The arithmetic series (7.8) is the simplest example of th ...

7.4 Infinite series 203 EXAMPLE 7.7Find the sum of the first nterms of the series 2 1·1 51 + 13 1·1 71 + 14 1·1 91 +1- The gener ...

204 Chapter 7Sequences and series so that the geometric series is the expansion of the function in powers of x: (7.17) The serie ...

7.5 Tests of convergence 205 The sum of the infinite series is then the limit of the sequence of partial sums. On the other hand ...

206 Chapter 7Sequences and series in whichs n contains 2 n terms of which the first, and largest, is 1 22 np . Each sum s n is t ...

7.5 Tests of convergence 207 (ii) The exponential series The general term is so that and. Then and the series converges for all ...

208 Chapter 7Sequences and series Alternating series If the terms of the seriesa 1 1 + 1 a 2 1 + 1 a 3 1 +1-become progressively ...

7.6 MacLaurin and Taylor series 209 (ii) The coefficient of x n in the series isc n 1 = 1 (−3) n 2 n 2 . By the ratio test,|c n ...