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)−31Hence, in a signal processing application, for small phase offsets θ\thetaθ, show that the windowing function W(θ)W(\theta)W(θ) defined by
W(θ)=cos(5θ)+(1−32θ2)−13 W(\theta) = \cos(5\theta) + \left(1 - \frac{3}{2}\theta^2\right)^{-\frac{1}{3}} W(θ)=cos(5θ)+(1−23θ2)−31can be approximated by A+Bθ+Cθ2A + B\theta + C\theta^2A+Bθ+Cθ2, where AAA, BBB and CCC are constants to be found.
308 exam-style questions on OCR (MEI) A Level Maths 1.6 Sequences and Series, covering 1.6.1 Binomial expansion for positive integer n, 1.6.2 Factorial and combinations notation, 1.6.3 Binomial expansion for rational n (A-level only), 1.6.4 Binomial expansion of (a + bx)^n via factoring (A-level only), 1.6.5 Binomial approximating polynomials (A-level only), 1.6.6 What a sequence is; finite and infinite (A-level only), 1.6.7 Generate a sequence by formula or recurrence (A-level only), 1.6.8 A series as a sum of terms (A-level only), 1.6.9 Sigma notation (A-level only), 1.6.10 Increasing, decreasing and periodic sequences (A-level only), 1.6.11 Convergent and divergent sequences (A-level only), 1.6.12 Arithmetic sequences and series (A-level only), 1.6.13 Standard formulae for arithmetic series (A-level only), 1.6.14 Geometric sequences and series (A-level only), 1.6.15 Standard formulae for geometric series (A-level only), 1.6.16 Sum to infinity of a geometric series (A-level only), and 1.6.17 Sequences and series in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.