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8 marksSeries

CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 2 · Question 4(a)(i)

A function is defined as g(x) = x sin(x/2).

Obtain the Maclaurin series expansion for g up to the term in x⁴.

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Other parts of this question

  1. 4(a)(ii)Hence, estimate g(2).[2 marks]
  2. 4(b)(i)Express the nᵗʰ partial sum S_n of the series using sigma notation.[2 marks]
  3. 4(b)(ii)Hence, calculate S₂₀ - S₁₈.[1 mark]
  4. 4(b)(iii)Given that Σ(n=1 to ∞) (1/n²) converges, show that S_n diverges.[4 marks]
  5. 4(c)Use the method of induction to prove that Σ(r=1 to n) r(r-1) = n(n² - 1) / 3.[8 marks]

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