A-LevelFurther MathematicsSummation of series9231/12 · Oct/Nov 2024 · Q59 Marks
It is given that S_n = Σ_{r=1}^n u_r, where u_r = x^(f(r)) − x^(f(r+1)) and x > 0.
- (a)Find S_n in terms of n, x and the function f.[2]
- (b)Given that f(r) = lnr, find the set of values of x for which the infinite series u₁ + u₂ + u₃+ ... is convergent and give the sum to infinity when this exists.[3]
- (c)Given instead that f(r) = 2log_x r where x ≠ 1, use standard results from the List of formulae (MF19) to find Σ_n=1^N S_n in terms of N. Fully factorise your answer.[4]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.
- [M1] Writes at least three terms, including last.
- [A1]
- [B1] Accept [0<]x ≤1
- [B1] ∑u_r = 1 [for x<1]
- [B1] ∑u_r = 0 for x=1
- [M1] Uses x^(2logxr) = r^2
- [A1]
- [M1] Substitutes formulae from MF19.
- [A1]
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