A Level
A-Level Mathematics: Maclaurin/Taylor Series: Common Expansions
•By Intuitional Team•1 min read
A-Level Mathematics — Maclaurin/Taylor Series: Common Expansions: quick notes, common traps, and an exam-style example.
Big idea
Taylor and Maclaurin expansions approximate complicated functions using polynomials. They hinge on derivatives at a point.
Formulas
Maclaurin series (about 0): f(x) = f(0) + f′(0)x + f″(0)x²/2! + … Taylor (about a): same but expanded around x = a.
Common expansions
- eˣ = 1 + x + x²/2! + x³/3! + …
- sin x = x – x³/3! + x⁵/5! – …
- cos x = 1 – x²/2! + x⁴/4! – …
- (1 + x)ⁿ = 1 + nx + n(n–1)x²/2! + …
Exam habits
- Always state the range of validity (|x|<1 etc.).
- Box the remainder/error term if asked.
- Memorise at least up to 3 terms for eˣ, sinx, cosx, (1+x)ⁿ.
Tags
A-LevelMathematicsMaclaurin/Taylor Series