Maths with Lemon

Maclaurin Series

“[On knowing he was gay from a young age] Once I was out of my mum, I never looked back.”

Colin Maclaurin

1. What Is a Maclaurin Series?

Prior knowledge

  • Differentiation and higher derivatives.
  • Factorials and sigma notation.
  • Infinite binomial expansions.
  • Polynomial approximation near a point.

General formula

\[f(x)=f(0)+xf^{\prime}(0)+\frac{x^2}{2!}f^{\prime\prime}(0)+\cdots\]
\[\boxed{f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n}\]

A Maclaurin series is a Taylor series centred at \(x=0\). The series represents the function where it converges to that function.

2. Deriving the Coefficients

Build the pattern

  • Assume a power series \(a_0+a_1x+a_2x^2+\cdots\).
  • Set \(x=0\) to find \(a_0\).
  • Differentiate and set \(x=0\) to find the next coefficient.
  • Continue to obtain the general term.

Coefficients

\[a_0=f(0),\qquad a_1=f^{\prime}(0)\]
\[a_2=\frac{f^{\prime\prime}(0)}{2!},\qquad a_3=\frac{f^{(3)}(0)}{3!}\]
\[\boxed{a_n=\frac{f^{(n)}(0)}{n!}}\]
Example: first four terms of \(e^x\)
\[\boxed{e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots}\]

3. Standard Series to Recognise

Essential AA HL expansions

  • The exponential function.
  • Sine and cosine (angles in radians).
  • Natural logarithms and inverse tangent.
  • The generalized binomial series.
  • Know the convergence restrictions when substituting into a series.

Formula collection

\[e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots\]
\[\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots\]
\[\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots\]
\[\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots\]
\[\arctan x=x-\frac{x^3}{3}+\frac{x^5}{5}-\cdots\]
\[(1+x)^p=1+px+\frac{p(p-1)}{2!}x^2+\cdots\]

For \(\ln(1+x)\) and \(\arctan x\), the series converge for \(-1<x\leq1\) and \(|x|\leq1\), respectively. The binomial expansion for non-negative integer \(p\) terminates; for other rational \(p\), use \(|x|<1\).

4. Substitution into Known Series

Method

  • Start from a known expansion.
  • Replace every \(x\) with the new expression.
  • Simplify the powers carefully.
  • Stop at the requested power of \(x\), even when some powers are absent.

Worksheet examples

Expand \(\sin(2x^2)\) through \(x^6\)
\[\sin u=u-\frac{u^3}{6}+\cdots\]
\[\boxed{\sin(2x^2)=2x^2-\frac{4}{3}x^6+O(x^{10})}\]
Expand \(e^{2x^2}\) through \(x^6\)
\[\boxed{e^{2x^2}=1+2x^2+2x^4+\frac43x^6+O(x^8)}\]

5. Multiplying and Dividing Series

What you need to know

  • Expand each factor to sufficient order.
  • Multiply and collect like powers.
  • Ignore terms beyond the requested degree.
  • For quotients, use a geometric/binomial series or coefficient comparison.

Worked example from the worksheet

Expand \(e^x\ln(1+x)\) through \(x^2\)
\[e^x=1+x+\frac{x^2}{2}+\cdots\]
\[\ln(1+x)=x-\frac{x^2}{2}+\cdots\]
\[\boxed{e^x\ln(1+x)=x+\frac{x^2}{2}+O(x^3)}\]
Further example: \(\frac{e^x}{1+x}\) through \(x^2\)
\[\boxed{\frac{e^x}{1+x}=1+\frac{x^2}{2}+O(x^3)}\]

6. Approximation and Convergence

Interpret the graphs

  • More terms generally improve the approximation near \(x=0\).
  • Accuracy away from zero depends on the function and its convergence interval.
  • A finite polynomial may behave very differently from the original function far from zero.
  • The Maclaurin series for \(\ln(1+x)\) does not converge at \(x=3\).

Compare the approximations

Estimate \(e^{0.2}\) using four terms
\[e^{0.2}\approx1+0.2+\frac{0.2^2}{2}+\frac{0.2^3}{6}\]
\[\boxed{e^{0.2}\approx1.22133}\]

The exact value is approximately \(1.22140\).

Use the interactive explorer below to see the approximation improve.

7. Interactive Maclaurin Explorer

Explore polynomial approximations

  • Choose \(e^x\), \(\sin x\), or \(\cos x\).
  • Move the degree slider to include more terms.
  • Compare the original function with its Maclaurin polynomial.
  • Observe where the approximation starts to diverge.

Original function vs polynomial

Blue: original function. Orange: Maclaurin polynomial.

8. Limits and Integration with Series

Two important applications

  • Use a truncated series to evaluate a limit near zero.
  • Integrate a polynomial approximation term by term.
  • Choose enough terms to achieve the requested precision.
  • Be careful with the range of validity.

Worked examples

Evaluate a limit
\[\cos x=1-\frac{x^2}{2}+O(x^4)\]
\[\boxed{\lim_{x\to0}\frac{\cos x-1}{x^2}=-\frac12}\]
Approximate an integral
\[e^{-x^2}=1-x^2+\frac{x^4}{2}+\cdots\]
\[\int_0^{1/2}e^{-x^2}\,dx\approx\left[x-\frac{x^3}{3}+\frac{x^5}{10}\right]_0^{1/2}\]
\[\boxed{\int_0^{1/2}e^{-x^2}\,dx\approx0.46146}\]

9. Maclaurin Series from Differential Equations

HL technique

  • Start with a differential equation and initial conditions.
  • Evaluate \(y(0)\), \(y^{\prime}(0)\), \(y^{\prime\prime}(0)\), and so on.
  • Differentiate the equation to obtain higher derivatives.
  • Insert these values into the Maclaurin formula.

Worked example from the worksheet

Solve approximately: \(y^{\prime}=x-y^2\), \(y(0)=1\)
\[y^{\prime}(0)=-1\]
\[y^{\prime\prime}=1-2yy^{\prime},\quad y^{\prime\prime}(0)=3\]
\[y^{(3)}=-2(y^{\prime})^2-2yy^{\prime\prime},\quad y^{(3)}(0)=-8\]
\[\boxed{y(x)\approx1-x+\frac32x^2-\frac43x^3}\]

This is the four-term Maclaurin polynomial; its value at \(x=1\) is only a rough extrapolation.

10. HL Revision and Challenges

Based on the worksheet

  • Find the first three non-zero terms for \(\arcsin x\).
  • Use products to expand \(e^x\sin x\).
  • Find approximations to roots using truncated series.
  • Develop Maclaurin solutions for first- and second-order differential equations.
  • Try the extended problems on pages 12–14.

Common mistakes

  • Confusing the first three terms with the first three non-zero terms.
  • Forgetting factorial denominators.
  • Missing a power when substituting \(x^2\) or \(x^3\).
  • Ignoring convergence restrictions.
  • Using too few terms when multiplying two expansions.

Open the complete Lesson 10.5 worksheet

IB Past Paper Problems

Select a topic and unit, then click the button below to get a new problem.

Press the button to start!