Maclaurin Series (AA HL)
A Maclaurin series rewrites a function as an infinite polynomial built from its value and derivatives at \(x=0\), giving a local approximation that gets more accurate the more terms you keep. At AA HL this is used both to derive series directly and to evaluate limits that l'Hôpital's rule would otherwise handle. This page covers the key series, worked examples, and the mistakes that lose the most marks. It's part of the broader Series & Limits topic.
20 questions on this sub-topic.
The key series
Covered under IB syllabus reference AHL5.13: evaluating limits using l'Hôpital's rule or a Maclaurin series, for the indeterminate forms \(\tfrac00\) and \(\tfrac{\infty}{\infty}\). All four expansions below are given in the formula booklet.
General Maclaurin series
\(f(x)=f(0)+f'(0)x+\dfrac{f''(0)}{2!}x^2+\cdots\)
Build any series by finding \(f(0)\) and successive derivatives at \(0\).
Standard expansions
\(e^{x}=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots\)
\(\sin x=x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\cdots\)
\(\cos x=1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\cdots\)
Learn to recognise these rather than re-deriving them each time.
Want the full topic overview and GDC guidance for this area? See Series & Limits.
Worked examples
Find the Maclaurin series for \(e^{x}\) up to and including the term in \(x^3.\)
Worked solution
\(f(x) = e^x\), all derivatives are \(e^x\), so \(f^{(n)}(0)\) M1 \(= 1.\) A1
\(f(x) = \sum \dfrac{f^{(n)}(0)}{n!}x^n.\) M1 A1 So \(e^x = 1 + x + \dfrac{x^2}{2} + \dfrac{x^3}{6}.\) A1
Using the Maclaurin series for \(e^x\) and \(\cos x\), find the series for \(e^{x}\cos x\) up to the term in \(x^2.\)
Worked solution
\(e^x = 1 + x + \tfrac{x^2}{2} + \cdots,\ \cos x\) M1 \(= 1 - \tfrac{x^2}{2} + \cdots\) A1
\((1 + x + \tfrac{x^2}{2})(1 - \tfrac{x^2}{2})\) M1 \(= 1 + x + \tfrac{x^2}{2} - \tfrac{x^2}{2}\) A1 \(= 1 + x + \cdots\) A1
Write down the first three terms of the Maclaurin series for \(e^{x}.\)
Worked solution
\(e^{x} = 1 + x + \dfrac{x^2}{2!} + \cdots\) A1
\(= 1 + x + \dfrac{x^2}{2} + \cdots\) A1
Common mistakes
- Treating a Maclaurin series as exact. It's a local approximation valid near \(x=0\) - the further \(x\) is from \(0\), the worse a truncated series estimates the true value.
- Dropping or misplacing a factorial. The \(e^x\), \(\sin x\) and \(\cos x\) series divide by \(n!\), not \(n\) - writing \(\tfrac{x^3}{3}\) instead of \(\tfrac{x^3}{3!}=\tfrac{x^3}{6}\) is a very common slip.
- Not keeping enough terms when multiplying two series. If you truncate each series to the target power before multiplying, a term that only appears after multiplying two lower-order terms together can go missing.
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20 Maclaurin series questions, marked instantly like the real exam.
Quick answers
What is a Maclaurin series?
A Maclaurin series expresses a function as an infinite sum of terms built from its derivatives at \(x=0\): \(f(x) = f(0) + f'(0)x + \tfrac{f''(0)}{2!}x^2 + \cdots\). It is a local polynomial approximation, most accurate near \(x=0\).
Are the Maclaurin series for e^x, sin x and cos x in the formula booklet?
Yes - the general Maclaurin series and the specific expansions for \(e^x\), \(\sin x\) and \(\cos x\) are all given in the formula booklet, so you don't need to memorise them, but you do need to know how to use them.