Is Maclaurin Series in Paper 1 or Paper 2?

Topic 5.2 is Pure Mathematics and can appear in Paper 1 (100 marks) or Paper 2 Section A (40 marks). Questions often combine the standard expansions with substitution or product scenarios.

Do I need to derive the standard expansions from scratch, or can I quote them?

You may quote the standard Maclaurin expansions (\$e^x\$, \$\sin x\$, \$\cos x\$, \$\ln(1+x)\$, \$(1+x)^n\$) without re-deriving them from scratch. However, if the question says "use the Maclaurin theorem" or "show that", you must derive term-by-term using \$f^\{(k)\}(0)/k!\$.

How many terms should I include in a series expansion?

Include all non-zero terms up to the power specified in the question. If no power limit is given but approximation is the goal, keep terms up to and including the first two or three non-zero terms and state that remaining terms are of higher order.

Is there a formula sheet for H2 Maths Maclaurin series?

Yes - the "Formulas at a glance" section near the top lists the general Maclaurin expansion and all five standard series (\$e^x\$, \$\sin x\$, \$\cos x\$, \$\ln(1+x)\$, \$(1+x)^n\$) with their ranges of validity. These standard series are given in MF27, but you must apply the right substitution and adjust the validity condition yourself. ---

H2 Maths Maclaurin Series Formula Sheet | Standard Expansions

Study guide07 Oct 2025, 00:00 ZUpdated 12 Oct 2026

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Marcus Pang·Managing Director (Maths)

H2 Maths Maclaurin series formula sheet: the general Maclaurin expansion, all five standard series expansions with ranges of validity, and approximation and error rules - aligne...

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Q: What does H2 Maths Notes (JC 1-2): 5.2) Maclaurin Series cover?
A: Series derivation, standard expansions, and approximation error handling for H2 Maths Topic 5.2.

Download: Get the H2 Maths Maclaurin Series formula sheet (PDF) for quick revision, or the complete notes (PDF) for the full walkthrough.

Before you revise
Memorise the first four terms of the classic expansions ( $e^x \space \sin x \space \cos x \space \ln(1 + x) \space (1 + x)^n$

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Series	Expansion	Valid for
$e^x$	$1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \dots$	all $x$
$\sin x$	$x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dots$
$\cos x$	$1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \dots$
$\ln(1 + x)$	$x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dots$
$(1 + x)^n$	$1 + nx + \dfrac{n(n - 1)}{2!}x^2 + \dots$

Technique	Rule
Substitution validity	Apply the base condition to the substituted input: $\ln(1+u)$ needs $\lvert u\rvert < 1$
Remainder estimate	For small $\lvert x\rvert$ , error $\approx$ the first omitted term's magnitude
Alternating series	Error $\lt$ next term

Question cue	Best first move	Why it works	Common trap
"Use Maclaurin's theorem" or "derive"	Find $f(0)$ , $f'(0)$ , $f''(0)$ , and higher derivatives as needed.	The coefficients must come from $f^{(r)}(0)/r!$ .	Quoting a memorised standard expansion without showing derivatives.
Function matches a standard form after replacing $x$	Set the inside expression as the new input, such as $u = 2x$ or $u = -x^2$ .	The standard template is still valid after substitution if the powers and range are updated.
Product or quotient of familiar functions	Write each needed series only up to powers that can affect the answer.	Higher powers cannot contribute to lower-order coefficients.	Expanding too many terms, then losing the requested order.
Decimal approximation near 0	Build the truncated polynomial first, then substitute the number and estimate the next omitted term.	The polynomial shows the approximation used and the next term gives error language.	Substituting the decimal before deciding how many terms are needed.

Target term	Pairs that contribute	Coefficient
constant	$1 \times 1$	$1$
$x$	$(2x) \times 1$	$2$
$x^2$	$(2x^2) \times 1$ and $1 \times \left(-\dfrac{x^2}{2}\right)$
$x^3$	$\left(\dfrac{4}{3}x^3\right) \times 1$

Series used	Substitution	Validity check	Common trap
$\ln(1+u)$ , valid for $\lvert u\rvert < 1$ in most approximation questions	$u = 2x$	Need $\lvert 2x\rvert < 1$ , so $\lvert x\rvert < \dfrac{1}{2}$	Copying $\lvert x\rvert < 1$ without changing it.
$(1+u)^n$ , valid for $\lvert u\rvert < 1$ when $n$ is not a non-negative integer	$u = -3x$
$\ln(1+u)$	$u = x^2$	Need $\lvert x^2\rvert < 1$

H2 Maths Maclaurin Series Formula Sheet | Standard Expansions

Formulas at a glance

Definition

Standard Expansions

Building New Series

Expansion route-choice checkpoint

Product coefficient checkpoint

Approximations and Error Bounds

Validity after substitution checkpoint

Solving Equations with Series

Truncated-equation root checkpoint

Calculators and Verification

Exam Watch Points

Practice Quiz

Quick Revision Checklist

Common exam mistakes

Frequently asked questions

Other H2 Maths formula sheets

Sources

Step	What to check	Why it matters	Common trap
1	Identify the expansion point and the small variable.	A Maclaurin approximation is built around $x = 0$ .	Treating every algebraic root as equally valid.
2	Note the highest power kept and the first omitted term.	The omitted term should be small at the candidate root.	Keeping a root where the omitted term is comparable to the retained terms.
3	Substitute each candidate root into the original equation if possible.	The original equation, not the truncated one, decides the true solution.	Reporting a root that only solves the approximation.
4	State whether the answer is a near-zero estimate or a numerical root needing another method.	This protects method marks and interpretation.	Presenting a far-from-zero root as a Maclaurin result.