A-LevelFurther MathematicsSummation of series9231/12 · Oct/Nov 2020 · Q37 Marks

By simplifying (xⁿ - √x²ⁿ + 1)(xⁿ + √x²ⁿ + 1), show that (1 / (xⁿ - √x²ⁿ + 1)) = -xⁿ - √x²ⁿ + 1.

(a)By simplifying (xⁿ - √x²ⁿ + 1)(xⁿ + √x²ⁿ + 1), show that (1 / (xⁿ - √x²ⁿ + 1)) = -xⁿ - √x²ⁿ + 1.[1]
(b)Let u_n = xⁿ⁺¹ + √x²ⁿ⁺² + 1 + (1 / (xⁿ - √x²ⁿ + 1)). Use the method of differences to find Σ(from n=1 to N) u_n in terms of N and x.[3]
(c)Deduce 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]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

[B1] Uses the difference of two squares and obtains the given answer. AG (xⁿ −√(x²ⁿ +1))(xⁿ + √(x²ⁿ +1)) = x²ⁿ −(x²ⁿ +1) = -1
[B1] Uses the identity given in part (a). U_n = xⁿ⁺¹ + √(x²ⁿ⁺² +1)-xⁿ - √(x²ⁿ+1)
[M1] Writes at least three complete correct terms, including the first and last. ΣU_n = x² + √(x⁴+1)−x−√(x²+1) + x³ + √(x⁶+1)−x²−√(x⁴+1) + ... + x^(N+1) + √(x²ᴺ⁺²+1)−xᴺ−√(x²ᴺ+1)
[A1] (Don't allow in terms of n.) = x^(N+1) + √(x²ᴺ⁺²+1)-x-√(x²+1)
[B1] -1<x<1
[M1 A1] Finds sum to infinity. x^(N+1) → 0 as n → ∞ ⇒ u₁ + u₂ + u₃ + ... = 1 − x −√(x² + 1)

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