A-LevelFurther MathematicsSummation of series9231/12 · May/June 2012 · Q36 Marks
Given that f(r) = 1 / ((r + 1)(r + 2)), show that
- f(r-1) - f(r) = 2 / (r(r + 1)(r + 2))[2]
- Hence find Σ (from r=1 to n) 1 / (r(r + 1)(r + 2))[3]
- Deduce the value of Σ (from r=1 to ∞) 1 / (r(r + 1)(r + 2))[1]
Mark scheme
Marking points from the official mark scheme (May/June 2012). The mark or mark type is shown in brackets.
- [M1] Proves initial result. f(r-1) - f(r) = 1/(r(r+1)) - 1/((r+1)(r+2))
- [A1] = (r+2-r)/(r(r+1)(r+2)) = 2/(r(r+1)(r+2)) (AG)
- [M1A1] Sets up method of differences. Σ 1/(r(r+1)(r+2)) = 1/2 [ (1/(1x2)) - (1/(n+1)(n+2)) ]
- [A1] Shows cancellation to get result. = 1/4 - 1/(2(n+1)(n+2)) (OE)
- [A1√] States sum to infinity. Σ(from r=1 to ∞) 1/(r(r+1)(r+2)) = 1/4
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