What Is a Maclaurin Series and How Do You Use It to Approximate a Function?

A Maclaurin series represents a function as an infinite polynomial centred at $x=0$. To approximate a function, truncate this series after a finite number of terms and substitute the required value of $x$.

The Reasoning and Method

Provided the series converges to the function, its Maclaurin expansion is

$$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n =f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3+\cdots.$$

Each coefficient is determined by a derivative evaluated at $x=0$. A finite polynomial formed from the first terms is called a truncated Maclaurin series or Maclaurin polynomial.

To construct a series directly, evaluate $f(0)$, $f'(0)$, and successive derivatives at zero. Then place each result over (n!) as the coefficient of $x^n$. This works because matching these coefficients makes the polynomial have the same value and successive derivatives as at the centre. The degree of the truncation is the highest power retained.

When using a known expansion, transform the variable carefully before substituting a numerical value. Replacing $x$ with $2x$ in the exponential series changes every power: $e^{2x}=1+2x+\frac{(2x)^2}{2!}+\cdots$. State the highest power retained so the examiner can identify exactly where truncation occurred.

For example, approximate $e^{0.2}$ using terms up to and including $x^3$:

$$e^{0.2}\approx 1+0.2+\frac{0.2^2}{2}+\frac{0.2^3}{6}=1.22133\ldots$$

Thus $e^{0.2}\approx 1.2213$. The actual value is approximately $1.22140$, so the truncated series gives a close approximation. Accuracy is generally better when $x$ is closer to zero or when more terms are retained.

A common misconception is that every Maclaurin series equals its function everywhere. This is false: equality requires convergence to the function, while a truncation is normally only an approximation.

IB Exam Technique

This is IB Mathematics AA HL, AHL 5.19. On Paper 1, show the substitution or coefficient derivation exactly; on Papers 2 and 3, technology may check the approximation, but working is still required. If asked to show an expansion, calculate the necessary derivatives or obtain it from a known series using substitution, multiplication, differentiation, or integration.