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CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2 · Question 3(a)

SECTION B (Module 2) Answer BOTH questions.

The sequence \{a_n\} is defined by a_1 = 1, a_{n+1} = 4 + 2\sqrt[3]{a_n}. Use mathematical induction to prove that 1 \le a_n \le 8 for all n in the set of positive integers.

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  1. 3(b)(i)a)Let k > 0 and let f(k) = \frac{1}{k^2}. Show that f(k) - f(k+1) = \frac{2k+1}{k^2(k+1)^2}.[3 marks]
  2. 3(b)(i)b)Show that \sum_{k=1}^n \left( \frac{1}{k^2} - \frac{1}{(k+1)^2} \right) = 1 - \frac{1}{(n+1)^2}.[5 marks]
  3. 3(b)(iii)Hence, or otherwise, prove that \sum_{k=1}^\infty \frac{2k+1}{k^2(k+1)^2} = 1.[3 marks]
  4. 3(c)(i)Obtain the first four non-zero terms of the Taylor Series expansion of \cos x in ascending powers of (x - \frac{\pi}{4}).[5 marks]
  5. 3(c)(ii)Hence, calculate an approximation to \cos(-\frac{\pi}{16}).[3 marks]

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