Series

You will be able to

• Expand (a + bx)ⁿ for a positive integer n using ⁿCᵣ or Pascal's triangle, simplifying each term ⁿCᵣ aⁿ⁻ʳ(bx)ʳ.
• Find the coefficient of a given power of x, and the term independent of x in (x + k/x)ⁿ, by using the general term and solving for r.
• Explain why the first few terms of an expansion approximate (a + bx)ⁿ when x is small, and find the sum of the coefficients by putting x = 1.

You will be able to

• Recognise an arithmetic progression (constant difference d) and a geometric progression (constant ratio r) from their terms.
• Calculate the nth term and the sum of the first n terms of an AP, uₙ = a + (n − 1)d and Sₙ = ½n[2a + (n − 1)d], and of a GP, uₙ = arⁿ⁻¹ and Sₙ = a(1 − rⁿ)/(1 − r).
• Determine the first term and the common difference or ratio from two given terms by forming and solving simultaneous equations.

You will be able to

• State that a geometric series has a sum to infinity only when |r| < 1, and explain why: the terms, and rⁿ, tend to zero.
• Calculate the sum to infinity S∞ = a/(1 − r), and use it to find a or r when S∞ is given.
• Determine how many terms are needed for Sₙ to be within a given distance of S∞, using the gap S∞ − Sₙ = arⁿ/(1 − r).