Determine the first two terms, in ascending powers of uuu, of the binomial expansion of
(1+34u)−13 \left(1 + \frac{3}{4}u\right)^{-\frac{1}{3}} (1+43u)−31A signal processing engineer models the gain G G\,G of a circuit at frequency offset x x\,x using the formula:
G(x)=cos(5x)+(1+6x2)−13 G(x) = \cos(5x) + \left(1 + 6x^2\right)^{-\frac{1}{3}} G(x)=cos(5x)+(1+6x2)−31Hence, for small values of xxx, show that the gain can be approximated by
G(x)≈A+Bx+Cx2 G(x) \approx A + Bx + Cx^2 G(x)≈A+Bx+Cx2where AAA, B B\,B and C C\,C are constants to be found.
308 exam-style questions on OCR A Level Maths 1.4 Sequences and Series, covering 1.4.1 Binomial expansion for positive integer n, 1.4.2 Link to binomial probabilities, 1.4.3 Binomial expansion for rational n (A-level only), 1.4.4 Validity of the expansion (A-level only), 1.4.5 Sequences (A-level only), 1.4.6 Increasing, decreasing and periodic sequences (A-level only), 1.4.7 Sigma notation (A-level only), 1.4.8 Arithmetic sequences and series (A-level only), 1.4.9 Geometric sequences and series (A-level only), 1.4.10 Sum to infinity of a geometric series (A-level only), and 1.4.11 Modelling with sequences and series (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.