IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus
Maclaurin Series
Build and manipulate series near x=0 for approximation, limits and differential equations.
Learning goal
Use standard Maclaurin series and obtain related expansions by substitution, products, differentiation and integration.
Syllabus link
AA HL: AHL 5.19 · Topic 5 Calculus.
Big idea
A Maclaurin polynomial is a local model built from derivatives at zero; additional terms usually improve the approximation near zero.
Key relationship
\(f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n\)
Core idea
A Maclaurin polynomial is a local model built from derivatives at zero; additional terms usually improve the approximation near zero.
Use standard Maclaurin series and obtain related expansions by substitution, products, differentiation and integration.
Key results and method
- Start from a standard series or compute derivatives at zero.
- Substitute carefully and keep only the powers required.
- When multiplying series, decide the highest required power before expanding.
- Differentiate/integrate term by term when appropriate.
Worked example
Find the Maclaurin expansion of \(e^{2x}\) to \(x^3\).
- Use \(e^u=1+u+u^2/2!+u^3/3!+\cdots\).
- Set \(u=2x\) and simplify.
Practice
Try these questions before moving on.
- Use substitution to expand \(e^{x^2}\).
- Multiply two standard series and identify a zero coefficient.
- Integrate a geometric-type series to derive the expansion of \(\arctan x\).
- Use leading terms to evaluate a limit as \(x\to0\).
Tutorials and premium resources
The finalized printable handout for this topic is a premium Radford Mathematics resource and is not offered as a free website download.
Visit the Radford Mathematics Store →