D: Sequences and series flashcards

Topic 5: D: Sequences and series

General term of the binomial expansion of (a + b)ⁿ, n a positive integer

ⁿCᵣ aⁿ⁻ʳ bʳ, with ⁿCᵣ = n!/(r!(n − r)!). There are n + 1 terms. Bracket b completely, sign included.

Binomial expansion of (1 + x)ⁿ for rational n, and its validity

(1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + n(n − 1)(n − 2)x³/3! + …, valid for |x| < 1.

Expand √(4 − x) up to the term in x², and state the validity.

2(1 − x/4)^(1/2) = 2(1 − x/8 − x²/128 + …) = 2 − x/4 − x²/64 + … Valid for |x| < 4.

Arithmetic sequence: nth term and sum of n terms

uₙ = a + (n − 1)d Sₙ = (n/2)(2a + (n − 1)d) = (n/2)(a + l)

Geometric sequence: nth term, sum of n terms and sum to infinity

uₙ = arⁿ⁻¹ Sₙ = a(1 − rⁿ)/(1 − r), r ≠ 1 S∞ = a/(1 − r), only for |r| < 1

Σ r from r = 1 to n, and Σ r from 1 to 100

Σ r = n(n + 1)/2 (an arithmetic series). For n = 100: 100 × 101/2 = 5050.

Recurrence relation, and increasing, decreasing and periodic sequences

A recurrence gives each term from the previous one: uₙ₊₁ = f(uₙ) with a first term. Increasing: uₙ₊₁ > uₙ. Decreasing: uₙ₊₁ < uₙ. Periodic: the terms repeat in a cycle.

How do you find the limit of a convergent recurrence uₙ₊₁ = f(uₙ)?

Set uₙ₊₁ = uₙ = L and solve L = f(L). For uₙ₊₁ = 0.5uₙ + 3: L = 0.5L + 3, so L = 6.

Find the coefficient of x³ in (1 + 2x)⁵.

⁵C₃ × 2³ = 10 × 8 = 80.