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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn9231Paper 2 · Further Pure Mathematics 2DifferentiationMaclaurin series

Differentiation · Subtopic 2 of 2

Maclaurin Series

Any smooth function can be approximated near $0$ by a polynomial whose derivatives match it there: $f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\cdots$ . Standard series for $e^x$ , $\sin x$ , $\cos x$ , $\ln(1+x)$ and $(1+x)^n$ combine by substitution, multiplication and differentiation.

Cambridge 9231 Paper 2Edexcel YFM01 WFM02Edexcel 9FM0 Paper 1Edexcel 9FM0 Paper 2AQA 7367 Paper 1AQA 7367 Paper 2≈ 45 minRevision 1 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Derive a Maclaurin series by repeated differentiation.
Recall and combine standard series.
State ranges of validity.
Use series for approximations and limits.

Pause and think

How does a calculator find $\sin0.3$ ? It can only add and multiply.

Section 2 of 14

Core idea

Start hereHow a calculator knows sin 0.3

A chip can only add and multiply. Yet it gives $\sin0.3=0.2955202\ldots$ instantly. It uses a polynomial built so that its value, slope, curvature and every higher derivative match $\sin x$ at $0$ : a Maclaurin series.

The big idea

$f(x)=\sum\frac{f^{(r)}(0)}{r!}x^r$ . Standard series for $e^x$ , $\sin x$ , $\cos x$ , $\ln(1+x)$ , $(1+x)^n$ combine by substitution and multiplication.

1. Deriving the series

Matching derivatives at 0

$f(x)=a_0+a_1x+a_2x^2+a_3x^3+\cdots$
Assume a power series.
$f(0)=a_0$
Put $x=0$ .
$f'(0)=a_1,\ f''(0)=2a_2,\ f'''(0)=3!\,a_3$
Differentiate and put $x=0$ each time.
$a_r=\frac{f^{(r)}(0)}{r!}$
General pattern.

Worked example $\ln(1+x)$

First three terms.

Goal: Series.
Strategy: Differentiate repeatedly.

$f(0)=0,\ f'=\frac1{1+x}\to1$
First two.
$f''=-\frac1{(1+x)^2}\to-1,\ f'''=\frac2{(1+x)^3}\to2$
Next two.
$x-\frac{x^2}2+\frac{2x^3}{6}=x-\frac{x^2}2+\frac{x^3}3$
Divide by factorials.

Answer

x-\frac{x^2}2+\frac{x^3}3-\cdots

, valid for

-1<x\le1

✓ CheckDifferentiating gives

1-x+x^2-\cdots=\frac1{1+x}

✓.

2. Combining series

Worked example $e^x\sin x$ to $x^3$

Expand.

Goal: Series.
Strategy: Multiply truncated series; collect.

$(1+x+\tfrac{x^2}2+\tfrac{x^3}6)(x-\tfrac{x^3}6)$
Standard series.
$x^2:\ 1\cdot1=1;\quad x^3:\ \tfrac12-\tfrac16=\tfrac13$
Pairs of terms.

Answer

x+x^2+\frac13x^3+\cdots

✓ CheckAt

x=0.1

0.1+0.01+0.000333=0.110333

; exact

e^{0.1}\sin0.1=1.10517\times0.099833=0.110333

✓.

Common misconceptionValidity

\ln(1+2x)

is valid only for

-\frac12<x\le\frac12

3. Limits and approximations

Worked example $\lim_{x\to0}\frac{x-\sin x}{x^3}$

Evaluate.

Goal: Limit.
Strategy: Series for $\sin x$ .

$x-\sin x=\tfrac{x^3}6-\tfrac{x^5}{120}+\cdots$
Substitute.
$\frac{x-\sin x}{x^3}=\tfrac16-\tfrac{x^2}{120}+\cdots\to\tfrac16$
Divide; let $x\to0$ .

Answer

\frac16

✓ Check

x=0.01

\frac{0.01-0.0099998333}{10^{-6}}=0.16667

✓.

Summary

$a_r=\frac{f^{(r)}(0)}{r!}$ .
Substitute into standard series.
State the range of validity.

Maclaurin series: Power series about $x=0$ .

HistoryColin Maclaurin. The Scottish mathematician Colin Maclaurin used these series in his 1742 "Treatise of Fluxions", though Taylor (1715) and earlier Gregory and Madhava knew special cases.

Fun factEuler's identity. Putting

ix

into the series for

e^x

splits it into the series for

\cos x

and

i\sin x

e^{ix}=\cos x+i\sin x

, so

e^{i\pi}+1=0

Section 3 of 14

Key formulae

Maclaurin

f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3+\cdots

Standard series

e^x=\sum\frac{x^r}{r!},\ \sin x=x-\frac{x^3}{3!}+\cdots,\ \cos x=1-\frac{x^2}{2!}+\cdots

Logarithm

\ln(1+x)=x-\frac{x^2}2+\frac{x^3}3-\cdots,\quad-1<x\le1

TrickSubstitute, don't differentiate. For

e^{3x}

\cos(x^2)

, put

3x

x^2

into the standard series. Differentiating from scratch is slower and riskier.

Section 4 of 14

Worked examples

Example 1 · A product0 / 3 steps

Find the series for

e^x\sin x

up to

x^3

Example 2 · A limit0 / 2 steps

Find

\lim_{x\to0}\frac{x-\sin x}{x^3}

Section 5 of 14

Explore the graph

Polynomials hugging sin x

Loading the interactive graph…

Try this

Each extra term matches $\sin x$ over a wider range.
Near $0$ even $y=x$ is an excellent approximation.

Maclaurin polynomials of degree 1, 3, 5.

In the real worldSmall-angle physics. The pendulum formula

T=2\pi\sqrt{l/g}

uses

\sin\theta\approx\theta

, the first term of the series; the next term gives the correction for wider swings.

Section 6 of 14

Common mistakes

Incorrect

f''(0)x^2

Correct

\frac{f''(0)}{2!}x^2

Each term divides by the factorial.

Incorrect

\ln x

has a Maclaurin series

CorrectUse

\ln(1+x)

\ln x

isn't defined at

0

IncorrectIgnoring validity

CorrectState the range

\ln(1+2x)

needs

-\frac12<x\le\frac12

Calculator: computing sin(0.3)

Secretly: 0.3 − 0.0045 + 0.00002…

Section 7 of 14

Exam tip

Exam tipCollect carefully. When multiplying series, list every pair of terms that gives the power you want before adding.

Section 8 of 14

Guided practice

SA-FP-MAC-G012 marksL1 FoundationNon-calculatorStandard series

Find the coefficient of

x^2

in the Maclaurin series of

e^{3x}

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-FP-MAC-0011 markL1 FoundationNon-calculatorStandard series

State the coefficient of

x^3

in the Maclaurin series of

\sin x

Salam Academy Original Practice · Proposed mark scheme

SA-FP-MAC-0022 marksL2 StandardNon-calculatorDeriving

Find the coefficient of

x^3

in the series for

\ln(1+x)

by differentiation (

f'''(0)/3!

Salam Academy Original Practice · Proposed mark scheme

SA-FP-MAC-0033 marksL3 ExamNon-calculatorStandard series

Find the coefficient of

x^3

in the series for

e^x\sin x

Salam Academy Original Practice · Proposed mark scheme

SA-FP-MAC-0042 marksL2 StandardNon-calculatorApplications

Use

\cos x\approx1-\frac{x^2}2

to estimate

\cos0.2

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-FP-MAC-0053 marksL4 ChallengingNon-calculatorApplications

Use series to find

\displaystyle\lim_{x\to0}\frac{x-\sin x}{x^3}

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Match derivatives at $0$ ; divide by $r!$ .
Substitute into standard series.
State validity.
Series give limits and approximations.

Section 12 of 14

Quick quiz

Q1 · SA-FP-MAC-Q011 markL1 FoundationNon-calculatorDeriving

The

x^n

coefficient in a Maclaurin series is

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: 9231 AS & A Level Further Mathematics
Where: Paper 2 · 2.5 Differentiation (Maclaurin series)
Expected: Derive the first few terms of a Maclaurin series by repeated differentiation, including for implicit functions.
Calculator: Permitted on all papers; exact answers are expected where the question asks.
Typical style: Typical: "find the first three non-zero terms" then "hence find an approximation for …".
Open in: 9231 · Paper 2

Section 14 of 14

Past paper questions

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Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Cambridge International 9231, Current syllabus (official specification).