A-LevelFurther MathematicsSummation of series9231/12 · Oct/Nov 2025 · Q19 Marks

Use standard results from the list of formulae (MF19) to find the following sums.

(a)Use standard results from the list of formulae (MF19) to find Σ (from r=1 to n) (r³-r) in terms of n, fully factorising your answer.[3]
(b)Express (r+3)/(r³-r) in the form A/(r-1) + B/r + C/(r+1) where A, B and C are constants to be determined, and hence use the method of differences to find Σ (from r=2 to n) (r+3)/(r³-r)[5]
(c)Deduce the value of Σ (from r=2 to ∞) (r+3)/(r³-r)[1]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.

[M1 A1] Substitutes formulae. For the 1st A mark allow any letter for n.
[A1] Factorises.
[M1 A1] Finds partial fractions. M mark is for an equation that would lead to values for A, B and C.
[M1 A1] Shows five complete terms including first two and last two.
[A1] SCB1 if insufficient terms to show all of the cancelling, but must start at r = 2
[B1FT] FT from their part (b). Needs at least two terms in part (b) with one involving a number and a term in n.

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