Key Idea: A Maclaurin series writes a function as a sum of powers of x. You build series, use known ones, and find limits with them. A Maclaurin part appears most years on Paper 1 or Paper 2.
Building a Maclaurin series
- eˣ: every power.
- sin x: odd powers only.
- cos x: even powers, 1 − x²/2 + x⁴/24 − …
- ln(1 + x): plain denominators, no factorials.
Using known Maclaurin series
- Substitute — Put an expression into a known series, e.g. x² into eˣ.
- Multiply — Expand two series and keep only the powers you need.
- Limit — Replace by the series, cancel, divide by the matching power, let x → 0.
Substituting into a series you already know is faster than differentiating.
Why does cos x have only even powers? Its odd derivatives are 0 at x = 0.
Limit of (eˣ − 1 − x)/x² as x → 0? 1/2.
Series for eˣ²? Put x² into the eˣ series: 1 + x² + x⁴/2 + …
Coefficient of x⁴ in cos x? 1/4! = 1/24.