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H2 Maths Notes 5 2 Maclaurin Series

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H2 Maths Notes (JC 1-2): 5.2) Maclaurin Series

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07 Oct 2025, 00:00 Z

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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.

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

Part 19 of 32

Previous topicH2 Maths Notes (JC 1-2): 5.1) Differentiation Next topicH2 Maths Notes (JC 1-2): 5.3) Integration Techniques

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). Practise deriving them quickly so you can adapt to composite functions and substitution questions.

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.

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

Start by substituting $e^{2x} \cos x$ into the $e^x$ template to obtain $1 + 2x + 2x^2 + \dfrac{4}{3} x^3 + \dots$ . Apply the same substitution to $\cos x$ to get $1 - \dfrac{x^2}{2} + \dfrac{x^4}{24} - \dots$ . Multiply the truncated polynomials and keep terms up to $x^3$ : $1 + 2x + \dfrac{3}{2} x^2 + \dfrac{1}{3} x^3 + \dots$ .

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 $\lvert x \rvert < 1$ ).

Example -- Approximate $\ln(1.08)$

Set $x = 0.08$ .
$\ln(1 + x) \approx x - \dfrac{x^2}{2} + \dfrac{x^3}{3} = 0.08 - 0.0032 + 0.0001707 = 0.07697$ (4 s.f.).
Error bounded by next term magnitude $\dfrac{x^4}{4} \approx 0.000010$ .

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.

Example -- Estimate solution

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

Expand $e^{-x} = 1 - x + \dfrac{x^2}{2} - \dfrac{x^3}{6} + \dots$ .
Equate: $1 - x + \dfrac{x^2}{2} \approx 1 - \dfrac{x}{2}$
Simplify: $-x + \dfrac{x^2}{2} = -\dfrac{x}{2}$
Solutions: $x = 0$ or $x = 1$ (valid for small positive root $x \approx 1$ ; improve by including cubic term).

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 $\lvert x \rvert < 1$ ).

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.

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

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H2 Maths Notes (JC 1-2): 5.2) Maclaurin Series

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-\dfrac{x}{2} + \dfrac{x^2}{2} = 0

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