H2 Maths Maclaurin Series Formula Sheet | Standard Expansions

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 ($e^x \space \sin x \space \cos x \space \ln(1 + x) \space (1 + x)^n$). Practise deriving them quickly so you can adapt to composite functions and substitution questions.

• Maclaurin means expand around $x=0$: Find values of derivatives at 0.
• Standard series are templates: Substitute carefully and keep only needed powers.
• Approximation needs range and error awareness: State validity and mention the next-term size when useful.

Concrete example: To expand $e^{2x}\cos x$ up to $x^3$, write both series first, multiply them, and discard terms above $x^3$.

Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 5.2 scope is within Section A Pure Mathematics, which is assessed in Paper 1 (100 marks) and Paper 2 Section A (40 marks).

Formulas at a glance

The general Maclaurin theorem, the five standard expansions you may quote, and their validity ranges - on one screen. All five standard series are given in MF27, but you must apply the right substitution and validity condition yourself. Worked examples appear in the sections below.

General Maclaurin expansion

$$f(x) = f(0) + f'(0)x + \dfrac{f''(0)}{2!}x^2 + \dfrac{f^{(3)}(0)}{3!}x^3 + \dots$$

Standard expansions

Approximation & error

Definition

Maclaurin series expands a differentiable function about $x = 0$:

$$f(x) = f(0) + f'(0) x + \dfrac{f''(0)}{2!} x^2 + \dfrac{f^{(3)}(0)}{3!} x^3 + \dots$$

Truncate after the required power; remainder term bounds the error.

Standard Expansions

$$\begin{aligned} e^x &= 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \dots \cr \sin x &= x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dots \cr \cos x &= 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \dots \cr \ln(1 + x) &= x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dots \cr (1 + x)^n &= 1 + nx + \dfrac{n(n - 1)}{2!} x^2 + \dfrac{n(n - 1)(n - 2)}{3!} x^3 + \dots \end{aligned}$$

Building New Series

• Substitute expressions: replace $x$ with $ax$ or $x^2$ to adapt templates.
• Multiply series: keep terms up to required power.
• For composite functions, write as product of known series and collect like powers.

Expansion route-choice checkpoint

Before expanding, decide whether the question is testing derivation, template adaptation, product collection, or approximation. The route decides what working earns the marks.

Worked check: for $(1 - 2x)^{-1}\sin x$ up to $x^3$, use the standard binomial series with $u=-2x$

Misconception check: "Maclaurin" does not always mean direct differentiation. Use derivatives when the question asks for them or when no standard template fits; otherwise, a clear adapted standard series is usually faster.

Example -- Expansion of $e^{2x} \cos x$

Start by substituting $x \mapsto 2x$ into the $e^x$ template to obtain $1 + 2x + 2x^2 + \dfrac{4}{3} x^3 + \dots$

Product coefficient checkpoint

When two series are multiplied, collect each power of $x$ by asking which term pairs have powers that add to the target power. For $e^{2x}\cos x$, keep only the terms that can affect up to $x^3$:

Common trap: do not multiply the first three displayed terms only and stop. A term like $x^3$ can come from a cubic term in one series, or from a linear term multiplied by a quadratic term in the other series.

Approximations and Error Bounds

• Remainder estimate: use next term magnitude when $\lvert x \rvert$ is small.
• For inequality bounds, apply alternating series test (error < next term).
• State approximation interval explicitly (e.g. valid for $\lvert x \rvert \lt 1$).

Validity after substitution checkpoint

When a standard series has a validity condition, apply the condition to the substituted expression, not just to $x$.

Worked check: for $\ln(1+3x)$, set $u=3x$. Since the base expansion uses $\lvert u\rvert < 1$, the substituted series is valid when $\lvert 3x\rvert < 1$

Misconception check: substituting into a known expansion changes both the terms and the validity interval.

Example -- Approximate $\ln(1.08)$

1. Set $x = 0.08$.
2. $\ln(1 + x) \approx x - \dfrac{x^2}{2} + \dfrac{x^3}{3} = 0.08 - 0.0032 + 0.0001707 = 0.07697$

Solving Equations with Series

• Replace functions with truncated series to solve equations near 0.
• Example: solve $e^x = 1 + kx$ for small $x$ by matching coefficients.

Truncated-equation root checkpoint

When a series is used to solve an equation, the algebra may produce roots that do not fit the "near 0" assumption. Check each candidate root against the size of the retained terms before accepting it.

Worked check: in the example below, the truncated equation gives $x = 0$ or $x = 1$. The root $x = 0$ is exactly at the Maclaurin centre. The root $x = 1$ is already far enough that omitted terms such as $x^3/6$

Misconception check: solving the truncated polynomial is not the same as solving the original equation. It is only a local approximation unless the question asks for an exact polynomial model.

Example -- Estimate solution

Solve $e^{-x} = 1 - \dfrac{x}{2}$ for small $x$.

1. Expand $e^{-x} = 1 - x + \dfrac{x^2}{2} - \dfrac{x^3}{6} + \dots$

Calculators and Verification

• Use graphing calculator (GC) series expansion (Series or taylor) to confirm terms before committing to final answers.
• Always write manual working; the GC is for checking only.
• When quoting decimal approximations, state the truncated polynomial clearly and show substitution.

Exam Watch Points

• Keep factorial denominators exact-do not evaluate unless simplifying later.
• State the remainder/order of approximation (e.g. “accurate up to $x^3$”).
• For composite functions, show substitution steps to avoid losing method marks.
• Mention validity range when required (usually $\lvert x \rvert \lt 1$).

Practice Quiz

Check that you can derive, manipulate, and apply Maclaurin expansions-including error language-without relying on memory aids.

Quick Revision Checklist

• Derive Maclaurin series directly from derivatives at zero.
• Memorise and adapt the standard expansions efficiently.
• Estimate errors using next-term bounds or alternating-series rules.
• Apply series to approximation or equation-solving problems with clear justifications.

Want weekly guided practice on Maclaurin Series? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.

Common exam mistakes

• Forgetting the factorial denominators: Writing $\sin x = x - x^3/3 + \dots$ instead of $x - x^3/3! + \dots$

Frequently asked questions

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 ($e^x$, $\sin x$, $\cos x$, $\ln(1+x)$, $(1+x)^n$

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 ($e^x$, $\sin x$, $\cos x$, $\ln(1+x)$, $(1+x)^n$

Other H2 Maths formula sheets

Revising more than one topic? Grab the matching one-page formula sheet:

• Functions & graphs: Functions · Graphs & Transformations · Equations & Inequalities
• Sequences & series: Sequences & Series
• Vectors: Vectors
• Calculus: Differentiation · Maclaurin Series (this page) · Integration Techniques · Definite Integrals · Differential Equations
• Statistics: Probability · Discrete Random Variables · Normal Distribution · Sampling · Hypothesis Testing · Correlation & Regression

For the official SEAB reference booklet, see the H2 Maths MF27 formula list.

Sources

• SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 5 Calculus sub-topic 5.2 Maclaurin series (expansions for $e^x$, $\ln(1+x)$, $\sin x$, $\cos x$, $(1+x)^n$

Next steps: Follow the H2 Maths notes hub into Topic 5.3 - Integration techniques to pair these expansions with area/volume work.