Maclaurin Series A-Level Further Maths Guide

A page of repeated derivatives can look less like a method and more like an endurance test. That is often the first obstacle with Maclaurin series A-Level Further Maths: students can follow the formula, but they are unsure what the calculation is trying to achieve.

The answer is simple. A Maclaurin series represents a function as an infinite power series centred at $x=0$, wherever that series converges to the function. Truncating it gives a polynomial approximation that can help with limits and equations.

Imagine a sixth-form student called Maya. She initially treats every expansion as a new formula to memorise. Her progress begins when she notices that the topic is built around one repeatable process: find derivatives, evaluate them at zero, and place the results into the series.

The Maclaurin series checklist

For most questions, use this sequence:

• Identify whether you should differentiate repeatedly or use a standard expansion.
• Decide the highest power of $x$ required.
• Substitute carefully into any standard series.
• Collect like powers of $x$ and discard unnecessary terms.
• State the general term or interval of validity when requested.
• Check the constant term by substituting $x=0$ into the original function.

That final check is surprisingly powerful. If $f(0)=3$, an expansion beginning with $1$ cannot be correct.

What is a Maclaurin series?

The Maclaurin series associated with a function that has derivatives of every order at $x=0$ is

$$f(0)+f^{\prime }(0)x+\frac{f^{\prime \prime }(0)}{2!}{x}^2+\frac{f^{\prime \prime \prime }(0)}{3!}{x}^3+\cdots .$$

Where this series converges to the function, we can write

$$f(x)=\sum _{r=0}^{\infty }\frac{f^{(r)}(0)}{r!}{x}^r.$$

The coefficient of $x^r$ is therefore determined by the $r$th derivative at zero. A Maclaurin series is a Taylor series with its centre fixed at $x=0$.

In an exam, you may be asked to find a fixed number of terms, continue up to a stated power, or provide a general term. Practise both expanded and sigma forms, checking your exam board's specification for its precise requirements.

Maclaurin series worked examples

Deriving the expansion of $e^{2x}$

Let

$$f(x)=e^{2x}.$$

Differentiate repeatedly:

$$f^{\prime }(x)=2e^{2x},f^{\prime \prime }(x)=2^2{e}^{2x},f^{\prime \prime \prime }(x)=2^3{e}^{2x}.$$

The pattern is

$$f^{(r)}(x)=2^r{e}^{2x}.$$

At $x=0$,

$$f^{(r)}(0)=2^r.$$

Substitution into the general formula gives

$$e^{2x}=\sum _{r=0}^{\infty }\frac{2^r}{r!}{x}^r=1+2x+\frac{2^2{x}^2}{2!}+\frac{2^3{x}^3}{3!}+\cdots .$$

Hence, up to and including the term in $x^3$,

$$e^{2x}=1+2x+2x^2+\frac{4}{3}{x}^3+\cdots .$$

Maya's useful observation is that repeated differentiation is not random here. Each derivative reveals the same multiplying pattern.

Using standard expansions for $e^x\cos x$

When a question asks you to use standard Maclaurin series, deriving everything again wastes time. Use

$$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\cdots$$

and

$$\cos x=1-\frac{x^2}{2}+\frac{x^4}{24}+\cdots .$$

Multiply the two series, retaining contributions up to $x^4$:

$$e^x\cos x=\left(1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\cdots \right)\left(1-\frac{x^2}{2}+\frac{x^4}{24}+\cdots \right).$$

Collecting like powers gives

$$e^x\cos x=1+x-\frac{x^3}{3}-\frac{x^4}{6}+\cdots .$$

Notice that the $x^2$ terms cancel. Cancellation is common in Maclaurin series exam questions, which is why organised working matters more than mental multiplication.

Open the A Level Further Maths Lessons and Question Bank when you want to repeat this method with guided questions rather than building every practice set yourself.

The standard series you should recognise

The most frequently used expansions include

$$e^x=\sum _{r=0}^{\infty }\frac{x^r}{r!},$$ $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots ,$$ $$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots ,$$

and

$$\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots .$$

The exponential, sine and cosine series converge for every real value of $x$. For the real logarithmic series shown above, the interval is

$$-1<x\le 1.$$

For a binomial expansion with a non-integer exponent or a negative integer exponent, convergence is guaranteed for $|x|<1$, although endpoint behaviour must be considered separately if a question asks for it.

How Maclaurin series produce approximations

A finite number of terms gives a polynomial approximation near $x=0$. For example,

$$\ln (1+x)\approx x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}.$$

Taking $x=0.1$ gives

$$\ln (1.1)\approx 0.1-\frac{0.1^2}{2}+\frac{0.1^3}{3}-\frac{0.1^4}{4}=0.095308\dots .$$

The approximation is effective because $0.1$ is close to the centre, $x=0$. In general, moving further from the centre can require more terms, and the value must remain inside the relevant interval of validity.

Choosing enough terms before calculating

The most useful planning happens before the algebra. Read the final expression and ask which early terms will disappear. In a limit question, subtraction may remove the constant and linear terms, leaving the quadratic term to determine the answer. Stopping before that term would leave you with no useful information.

For products, organise your working by the power you are collecting rather than multiplying everything indiscriminately. Each coefficient comes from pairs of terms whose powers add to the required power. This keeps the calculation short while making omissions easier to spot.

For substitutions, keep brackets around the whole expression being inserted. A multiplier inside a power affects the entire coefficient, not just the first factor. Simplify one term at a time before collecting the result.

Finally, distinguish an exact infinite series from a finite approximation. An equals sign is appropriate when the omitted terms are represented by an ellipsis and the series equals the function. When you give only a finite polynomial as a numerical approximation, use an approximation sign instead. That small distinction makes your mathematical argument clearer.

Exam-style practice

Try these without looking at the answers first.

• Find the expansion of $e^{-3x}$ up to and including the term in $x^3$.
• Expand $\ln (1+2x)$ up to and including the term in $x^4$, and state its interval of validity.
• Use a Maclaurin series to evaluate $\underset{x\to 0}{\lim }\frac{e^x-1-x}{x^2}$.

The answers are

$$e^{-3x}=1-3x+\frac{9}{2}{x}^2-\frac{9}{2}{x}^3+\cdots ,$$ $$\ln (1+2x)=2x-2x^2+\frac{8}{3}{x}^3-4x^4+\cdots ,-\frac{1}{2}<x\le \frac{1}{2},$$

and

$$\underset{x\to 0}{\lim }\frac{e^x-1-x}{x^2}=\frac{1}{2}.$$

For broader paper practice, use the OCR Further Maths Pure Core paper, this Edexcel Further Pure Mathematics paper and the Cambridge Further Pure Mathematics practice paper. You can also compare your presentation with the Cambridge specimen mark scheme. Choose papers that match your qualification and specification.

Common mistakes with Maclaurin series

Forgetting the factorial

The coefficient of $x^r$ is

$$\frac{f^{(r)}(0)}{r!},$$

not simply $f^{(r)}(0)$. Write the factorials into the formula before substituting values.

Stopping the inputs too early

If one series is substituted into another, higher powers can still contribute to the requested final order. Decide which input terms are needed before discarding anything.

Mixing up the power and the number of terms

An expansion through $x^4$ does not always contain five non-zero terms. For example, $\cos x$ has no terms in $x$ or $x^3$.

Ignoring the instruction

If a question says “hence” or asks you to use standard series, a fresh differentiation method may not receive all the intended method marks. Follow the route signalled by the question.

Omitting validity

A correct algebraic expansion can still be incomplete if the question requests the range of $x$. Apply any substitution to the original validity condition carefully.

Turn the method into exam confidence

Maya's turning point is not learning another trick. It is practising the same decision repeatedly: derive or substitute, choose the required order, collect powers, and check the result. Knowing that routine is different from carrying it out accurately under timed conditions.

Build that accuracy now with the Maclaurin series Lessons and Question Bank.