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CAPE Pure Mathematics Unit 2 · May/June 2022 · Paper 2 · Question 3(b)

Calculate the sum to infinity of the series Σ_{r=2}^{∞} 10/(r² - 1).

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  1. 3(a)(i)Use mathematical induction to prove that 3/4 + 5/36 + ... + (2n-1)/(n²(n-1)²) = 1 - 1/n², n ≥ 1.[9 marks]
  2. 3(a)(ii)Hence, or otherwise, calculate S₃₀ - S₁₀.[4 marks]
  3. 3(c)Determine the Taylor series expansion of x sin(x/2) about x = π, up to and including the first THREE non-zero terms.[5 marks]

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