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5 marksThe Binomial Theorem

CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2 · Question 4(a)(ii)

The series expansion of (1 + x)^k is given as 1 + kx + (k(k-1)x^2)/2! + (k(k-1)(k-2)x^3)/3! + (k(k-1)(k-2)(k-3)x^4)/4! + ... where k in R and -1 < x < 1.

Determine the series expansion of f up to and including the term in x^4.

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Other parts of this question

  1. 4(a)(i)Show that f(x) = (1 + 2x)^(1/3).[3 marks]
  2. 4(a)(iii)Hence, approximate f(0.4) correct to 2 decimal places.[3 marks]
  3. 4(b)(i)Show that h(x) = 0 has a root on the interval [0, 1].[3 marks]
  4. 4(b)(ii)Use the iteration x_(n+1) = 1 / (x_n^2 + 1) with initial estimate x_1 = 0.7 to estimate the root of h correct to 2 decimal places.[6 marks]
  5. 4(c)Use two iterations of the Newton-Raphson method with initial estimate x_1 = 1 to approximate the root of the equation g(x) = e^(4x - 3) - 4 in the interval [1,…[5 marks]

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