Maclaurin series and approximations (IB Math AA HL)

Key ideas

• The general series f(x) = f(0) + x f′(0) + x²/2! f″(0) + … and the series for eˣ, ln(1 + x), sin x, cos x and arctan x are in the formula booklet.
• Substitute to get new series: replace x by 2x, −x or x². Multiply two series and keep only the terms up to the power you need.
• Differentiate or integrate a series term by term (for example, integrate the series for 1/(1 + x²) to get arctan x).
• The series for ln(1 + x) is only valid for −1 < x ≤ 1.
• For a limit as x → 0, replace each function by its first few terms, cancel the lowest power of x, then let x → 0.

Worked example

Find the Maclaurin series for e²ˣ sin x up to the term in x³, and use it to find the limit of (e²ˣ sin x − x)/x² as x → 0.

1. Replace x by 2x in the series for eˣ: e²ˣ = 1 + 2x + 2x² + (4/3)x³ + ….
2. sin x = x − x³/6 + ….
3. Multiply, keeping powers up to x³: e²ˣ sin x = x + 2x² + 2x³ − x³/6 + … = x + 2x² + (11/6)x³ + ….
4. (e²ˣ sin x − x)/x² = (2x² + (11/6)x³ + …)/x² = 2 + (11/6)x + …, which tends to 2 as x → 0.

Answer: e²ˣ sin x = x + 2x² + (11/6)x³ + …, and the limit is 2.

Exam-style HL AA questions on this topic

• Differential Equations and Maclaurin questions — 62 questions with mark schemes, plus free worked examples
• All HL AA Calculus questions by topic

Maths IA ideas that use maclaurin series and approximations

• How does a calculator work out sin(x)?
• How many random numbers until the total passes 1? Finding e by chance
• Why does 1 + 1/4 + 1/9 + … equal π²/6?
• How many terms do you need? Fast and slow series for π
• All HL AA IA ideas

Related in Calculus

• Differential equations — separable and integrating factor
• Volumes of revolution and improper integrals
• Differentiation — implicit, logarithmic, and inverse-trig

Revise Maclaurin series and approximations for the exam

• HL AA revision notes — every unit summarised, with the key results to learn
• HL AA past papers — specimen papers and timed practice papers
• Formulas for Calculus — on the one-page HL AA formula sheet (free PDF), with the ones not in the IB booklet marked
• HL AA formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

TOK link: What role does intuition play in producing mathematical knowledge?

Stretch yourself: original calculus and functions problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S133. All 24 →