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A-LevelFurther MathematicsSummation of series9231/12 · Oct/Nov 2021 · Q37 Marks

Let S_n = Σ[from r=1 to n] ln( r(r+2) / (r+1)² ).

  1. (a)Using the method of differences, or otherwise, show that S_n = ln( (n+2) / (2(n+1)) ).[4]
  2. (b)Let S = Σ[from r=1 to ∞] ln( r(r+2) / (r+1)² ). Find the least value of n such that |S_n - S| < 0.01.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

  1. [B1] (a) Separates logarithms into correct form using a difference. Or as logarithm of product.
  2. [M1 A1] (a) Shows enough terms to make cancellation clear.
  3. [A1] (a) AG
  4. [B1] (b) States sum to infinity. AEF
  5. [M1] (b) Forms inequality ln((n+2)/(n+1)) < 0.01 leading to n+2 < e^0.01(n+1)
  6. [A1] (b) CAO

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About this question

Topic
Summation of series, A-Level Further Mathematics (syllabus 9231). Structured question, 7 marks.
Source
A-Level Further Mathematics, Oct/Nov 2021 Paper 1 Variant 2, question 3.

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