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CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2 · Question 4(a)(ii)b)

Hence, or otherwise, find the sum S_n = \sum_{r=1}^{n} \frac{r}{(r + 1)!}.

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  1. 4(a)(i)By expressing ^n\mathrm{C}_r and ^n\mathrm{C}_{r-1} in terms of factorials, prove that ^n\mathrm{C}_r + {}^n\mathrm{C}_{r-1} = {}^{n+1}\mathrm{C}_r.[6 marks]
  2. 4(a)(ii)a)Given that r is a positive integer and f(r) = \frac{1}{r!}, show that f(r) - f(r + 1) = \frac{r}{(r + 1)!}.[3 marks]
  3. 4(a)(ii)c)Deduce the sum to infinity of S_n in (ii) b) above.[2 marks]
  4. 4(b)(i)Show that the function f(x) = x^3 - 6x + 4 has a root x in the closed interval [0, 1].[5 marks]
  5. 4(b)(ii)By taking 0.6 as a first approximation of x_1 in the interval [0, 1], use the Newton-Raphson method to obtain a second approximation x_2 in the…[4 marks]

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