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

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.

Definition

Maclaurin series expands a differentiable function about \( x = 0 \):

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

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

Standard Expansions

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

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 \( x \mapsto 2x \) into the \( e^x \) template to obtain \( 1 + 2x + 2x^2 + frac{4}{3} x^3 + \dots \). Apply the same substitution to \( \cos x \) to get \( 1 - frac{x^2}{2} + frac{x^4}{24} - \dots \). Multiply the truncated polynomials and keep terms up to \( x^3 \): \( 1 + 2x + frac{3}{2} x^2 + frac{1}{3} x^3 + \dots \).

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 \( \ln(1.08) \)

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

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 \( e^{-x} = 1 - \tfrac{x}{2} \) for small \( x \).

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

Calculators and Verification

• Use GC series expansion (Series or taylor) to confirm terms before committing to final answers.
• Always write manual working; 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 \( \lvert x \rvert < 1 \)).

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: Proceed to Topic 5.3 for integration techniques that complement these series expansions.