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8 marksSequences

CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2 · Question 3(a)(ii)

A sequence {a_n} is defined by a_1 = sqrt(2) and a_(n+1) = sqrt(2 + a_n).

Use mathematical induction to prove that a_n is increasing and bounded above by 3, so a_n < a_(n+1) and a_n <= 3 for all n in N.

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Other parts of this question

  1. 3(a)(i)State the third term, a_3, of the sequence.[2 marks]
  2. 3(b)(i)Let f(x) = e^(-x^2). By calculating the first three non-zero terms and assuming the pattern continues, show that the Maclaurin series expansion of f(x) may be…[8 marks]
  3. 3(b)(ii)Hence, or otherwise, determine the values of x for which the expansion is valid.[3 marks]
  4. 3(c)Determine the sum of the series sum_(n=1)^infinity (sin(1/n) - sin(1/(n+1))).[4 marks]

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