A-LevelFurther MathematicsSummation of series9231/12 · May/June 2020 · Q410 Marks
By first expressing 1 r2-1 in partial fractions, show that n Σ r=2 1 r2-1 = 3 4 - an+b 2n(n+1) , where a and b are integers to be found.
- (a)By first expressing 1 r2-1 in partial fractions, show that n Σ r=2 1 r2-1 = 3 4 - an+b 2n(n+1) , where a and b are integers to be found.[5]
- (b)Deduce the value of ∞ Σ r=2 1 r2-1 .[1]
- (c)Find the limit, as n → ∞, of 2n Σ r=n+1 n r2-1 .[4]
Mark scheme
Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.
- [M1 A1] 1/(r²-1) = 1/((r-1)(r+1)) = 1/2 (1/(r-1) - 1/(r+1))
- [M1] Σ 1/(r²-1) = 1/2 [ (1/1 - 1/3) + (1/2 - 1/4) + (1/3 - 1/5) + ... + (1/(n-2) - 1/n) + (1/(n-1) - 1/(n+1)) ]
- [A1] = 1/2 [1 + 1/2 - 1/n - 1/(n+1)]
- [A1] = 3/4 - (2n+1)/(2n(n+1))
- [B1] 3/4
- [M1] Σ (from r=n+1 to 2n) n/(r²-1) = n [ Σ (from r=2 to 2n) 1/(r²-1) - Σ (from r=2 to n) 1/(r²-1) ]
- [M1A1] = n [ (3/4 - (4n+1)/(4n(2n+1))) - (3/4 - (2n+1)/(2n(n+1))) ]
- [A1] →½ as n → ∞
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