Introduction
The Maclaurin series is one of the most powerful tools in advanced calculus, and it appears in IB Math Analysis and Approaches HL. As a special case of the Taylor series, the Maclaurin series expands a function into an infinite polynomial centered at x = 0.
For IB students, mastering the Maclaurin series is essential for solving approximation problems, handling complex functions, and demonstrating higher-level mathematical reasoning.
Quick Start Checklist for Maclaurin Series
- Memorize the general Maclaurin series formula.
- Expand standard functions like eˣ, sin x, cos x, and ln(1 + x).
- Use partial sums to approximate values.
- Know the convergence conditions for each series.
- Practice IB HL exam-style questions using RevisionDojo resources.
The Maclaurin Series Formula
The Maclaurin series is the Taylor expansion around x = 0:
f(x) = f(0) + f′(0)x + (f″(0)/2!)x² + (f‴(0)/3!)x³ + …
Or in summation form:
f(x) = Σ [f⁽ⁿ⁾(0)/n!] xⁿ
Where f⁽ⁿ⁾(0) is the nth derivative evaluated at 0.
Standard Maclaurin Series Expansions
IB HL students should know these key expansions:
- eˣ = 1 + x + x²/2! + x³/3! + …
- sin x = x – x³/3! + x⁵/5! – …
- cos x = 1 – x²/2! + x⁴/4! – …
- ln(1 + x) = x – x²/2 + x³/3 – … (|x| < 1)
- (1 + x)ⁿ = 1 + nx + n(n – 1)x²/2! + …
