Maclaurin series
Objective. Derive local polynomial representations from derivatives.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A Maclaurin series represents a function f(x) as an infinite polynomial built from its derivatives evaluated at zero: f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ..., giving a local polynomial approximation around x=0. Each coefficient comes from successive differentiation and substitution of x=0, so the series is derived rather than guessed. For example, since e^x has every derivative equal to e^x, and e^0=1, the Maclaurin series for e^x is 1 + x + x²/2! + x³/3! + ....
Worked example
Find the Maclaurin series for f(x) = cos x up to and including the term in x⁴.
- 1f(0)=1, f'(x)=−sin x so f'(0)=0, f''(x)=−cos x so f''(0)=−1
- 2f'''(x)=sin x so f'''(0)=0, f''''(x)=cos x so f''''(0)=1
- 3Build the series: 1 + 0×x + (−1)x²/2! + 0×x³/3! + (1)x⁴/4!
- 4Simplify: 1 − x²/2 + x⁴/24
Answer: cos x ≈ 1 − x²/2 + x⁴/24
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Students forget to divide by n! when writing each term, treating the series as f(0)+f'(0)x+f''(0)x²+... without the factorial denominators. Always include the n! in the denominator of the nth term.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.