A-LevelFurther MathematicsSummation of series9231/12 · May/June 2013 · Q59 Marks
Use the method of differences to show that Σ_{r=1}^N 1/((2r + 1)(2r + 3)) = 1/6 - 1/(2(2N + 3)). Deduce that Σ_{r=N+1}^2N 1/((2r + 1)(2r + 3)) < 1/(8N).
- (a)Use the method of differences to show that Σ_{r=1}^N 1/((2r + 1)(2r + 3)) = 1/6 - 1/(2(2N + 3)).[5]
- (b)Deduce that Σ_{r=N+1}^2N 1/((2r + 1)(2r + 3)) < 1/(8N).[4]
Mark scheme
Marking points from the official mark scheme (May/June 2013). The mark or mark type is shown in brackets.
- M1A1 Finds partial fractions. 1/((2r+1)(2r+3)) = 1/2 * (1/(2r+1) - 1/(2r+3))
- M1A1 Expresses terms as differences. = 1/2 [(1/3 - 1/5) + ... + (1/(2N+1) - 1/(2N+3))]
- A1 Shows cancellation. = 1/2 [1/3 - 1/(2N+3)] (AG)
- M1 Uses Σ_{r=N+1}^2N = Σ_{r=1}^2N - Σ_{r=1}^N
- A1 Applies result = 1/2 [1/(2N+3) - 1/(4N+3)]
- M1 and simplifies. N/((2N+3)(4N+3))
- A1 Deduces inequality. < 1/(2N.4N) = 1/(8N) (AG)
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