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5 marksSequences

CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2 · Question 3(a)(i)

The nth term of a sequence is given by T_n = \frac{2n+1}{\sqrt{n^2+1}}.

Determine \lim_{n \to \infty} T_n.

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Other parts of this question

  1. 3(a)(ii)Show that T_4 = \frac{9}{4} \left(1 + \frac{1}{16}\right)^{-\frac{1}{2}}.[3 marks]
  2. 3(a)(iii)Hence, use the binomial expansion with x = \frac{1}{16} to approximate the value of T_4 for terms up to and including x^3. Give your answer correct to…[4 marks]
  3. 3(b)(i)Express the nth partial sum S_n of the series in sigma notation.[2 marks]
  4. 3(b)(ii)Hence, given that \sum_{n=1}^\infty \frac{1}{n^2} converges to \frac{\pi^2}{6}, show that S_n diverges as n \to \infty.[4 marks]
  5. 3(c)Use the method of mathematical induction to prove that \sum_{r=1}^n r(r-1) = \frac{n(n^2-1)}{3}.[7 marks]

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