Maclaurin and Taylor Series

Q: Do I need to memorise standard Maclaurin expansions for the exam?

No, all standard expansions (e^x, sin x, cos x, ln(1+x), arctan x) and their validity ranges are provided in the official FP2 formula booklet. However, you must be able to derive these from first principles when explicitly asked in a question.

Q: What is the difference between a Maclaurin and Taylor series?

A Maclaurin series is a special case of a Taylor series where the expansion is centred at $a=0$. Taylor series can be constructed for any centre point $a$.

Q: Can I use a calculator to compute derivatives for series questions?

Basic scientific calculators are permitted, but CAS-enabled calculators are banned. You must show full working for all derivative calculations to gain full marks, even if you use a calculator to check values.

Edexcel International A-Level Further Mathematics· FP2 Topic 6 (2018 Issue 3 Spec)· 25 min read

1. Higher Order Derivatives for Series Expansions★★☆☆☆⏱ 5 min

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📘 Definition

Higher Order Derivative

f^{(r)}(x)

The result of differentiating a function (f(x)) (r) times with respect to (x), where (r \geq 1)

Example:

The 3rd derivative of (\sin x) is (-\cos x)

To construct Maclaurin and Taylor series, you will need to compute derivatives of order 3 and above for simple functions. For composite functions, apply the chain rule repeatedly, and for products, use the product rule as required. You do not need to simplify factorial terms in series coefficients unless explicitly asked to do so.

📐 Worked Example

Find the first, second, third and fourth derivatives of (f(x) = \ln(1 + 2x)) evaluated at (x=0).

1
First derivative (chain rule):
$f'(x) = \frac{2}{1+2x} \implies f'(0) = 2$
2
Second derivative:
$f''(x) = -\frac{4}{(1+2x)^2} \implies f''(0) = -4$
3
Third derivative:
$f'''(x) = \frac{16}{(1+2x)^3} \implies f'''(0) = 16$
4
Fourth derivative:
$f^{(4)}(x) = -\frac{96}{(1+2x)^4} \implies f^{(4)}(0) = -96$

Exam tip:

Label derivatives clearly with their order when computing them for series questions, as markers award marks for correct derivative values even if you make a substitution error later.

2. Maclaurin Series Expansions★★★☆☆⏱ 7 min

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📘 Definition

Maclaurin Series

f(x) = \sum_{r=0}^{\infty} \frac{f^{(r)}(0)}{r!}x^r

A power series expansion of a function (f(x)) centred at (x=0), valid for values of (x) where the series converges

Example:

The Maclaurin series of (e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...)

When asked to derive a Maclaurin series from first principles, you must compute each required derivative explicitly, substitute (x=0) into each derivative, then plug the values into the general Maclaurin form. You can use standard expansions from the formula booklet for composite functions unless derivation is explicitly required.

📐 Worked Example

Derive the Maclaurin series expansion of (f(x) = \ln(1 + 2x)) up to and including the term in (x^4), and state its validity.

1
Use the derivative values from the previous worked example, plus the base function value:
$f(0) = \ln(1) = 0$
2
Substitute into the general Maclaurin series formula:
$f(x) = f(0) + x f'(0) + \frac{x^2}{2!}f''(0) + \frac{x^3}{3!}f'''(0) + \frac{x^4}{4!}f^{(4)}(0)$
3
Plug in derivative values:
$f(x) = 0 + 2x + \frac{x^2}{2}(-4) + \frac{x^3}{6}(16) + \frac{x^4}{24}(-96)$
4
Simplify terms:
$f(x) = 2x - 2x^2 + \frac{8}{3}x^3 - 4x^4$
5
Validity: Standard (\ln(1+u)) is valid for (|u| < 1), so (|2x| < 1 \implies |x| < 0.5

Exam tip:

Always state the validity range if asked, using standard ranges from the formula booklet for composite functions.

3. Taylor Series Expansions About a General Point★★★☆☆⏱ 7 min

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📘 Definition

Taylor Series (about point a)

f(x) = \sum_{r=0}^{\infty} \frac{f^{(r)}(a)}{r!}(x-a)^r

A power series expansion of a function (f(x)) centred at (x=a), used to approximate values of (f(x)) near (x=a)

Taylor series follow the same structure as Maclaurin series, except derivatives are evaluated at (x=a) instead of (x=0), and terms use powers of ((x-a)) instead of (x). They are used when you need an approximation of a function near a point other than 0.

📐 Worked Example

Find the Taylor series expansion of (f(x) = \sin x) in ascending powers of ((x - \pi)) up to and including the term in ((x - \pi)^3).

1
Compute derivatives evaluated at (a = \pi):
$f(\pi) = \sin \pi = 0, f'(\pi) = \cos \pi = -1, f''(\pi) = -\sin \pi = 0, f'''(\pi) = -\cos \pi = 1$
2
Substitute into the general Taylor series formula:
$f(x) = f(\pi) + (x-\pi)f'(\pi) + \frac{(x-\pi)^2}{2!}f''(\pi) + \frac{(x-\pi)^3}{3!}f'''(\pi)$
3
Plug in values:
$f(x) = 0 + (x-\pi)(-1) + \frac{(x-\pi)^2}{2}(0) + \frac{(x-\pi)^3}{6}(1)$
4
Simplify:
$f(x) = -(x-\pi) + \frac{1}{6}(x-\pi)^3$

Exam tip:

You do not need to expand ((x-a)) terms unless explicitly asked; leaving them in factored form is acceptable for full marks.

4. Series Solutions of Differential Equations★★★★☆⏱ 8 min

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To find a series solution of a differential equation, repeatedly differentiate the given ODE, use initial conditions to evaluate each derivative at the given point (almost always (x=0) for FP2 questions), then substitute the values into the Maclaurin series form for (y(x)).

📐 Worked Example

Find the series solution of (\frac{d^2y}{dx^2} + x \frac{dy}{dx} + y = 0) with initial conditions (y=1) and (\frac{dy}{dx}=0) at (x=0), up to and including the term in (x^4).

1
State initial values: (y(0)=1), (y'(0)=0). Rearrange ODE to isolate highest derivative:
$y'' = -x y' - y \implies y''(0) = -0 -1 = -1$
2
Differentiate ODE once to find third derivative:
$y''' = -y' - x y'' - y' = -x y'' - 2y' \implies y'''(0) = -0 - 0 = 0$
3
Differentiate again to find fourth derivative:
$y^{(4)} = -y'' - x y''' - 2y'' = -x y''' - 3y'' \implies y^{(4)}(0) = -0 - 3(-1) = 3$
4
Substitute into Maclaurin series for (y(x)):
$y = 1 + 0x + \frac{x^2}{2!}(-1) + \frac{x^3}{3!}(0) + \frac{x^4}{4!}(3)$
5
Simplify:
$y = 1 - \frac{1}{2}x^2 + \frac{1}{8}x^4$

Exam tip:

Show all steps of differentiating the ODE, as these are awarded method marks even if you make an arithmetic error when evaluating derivatives.

5. Common Pitfalls

Wrong move:

Using powers of (x) instead of ((x-a)) for Taylor series expansions about (a \neq 0)

Why:

Taylor series are centred at (a), so terms must be powers of the difference between (x) and (a)

Correct move:

Always use ((x-a)) as the base for all terms in a Taylor series expansion about (a)

Wrong move:

Quoting a standard expansion from the formula booklet when the question asks for derivation from first principles

Why:

Derivation questions require explicit demonstration of derivative calculation, not just recall of formulae

Correct move:

Compute all required derivatives explicitly, evaluate them at the centre point, and substitute into the general series formula for derivation questions

Wrong move:

Forgetting to divide each series term by the factorial of the derivative order

Why:

The general series formula includes a denominator of (r!) for the rth derivative term, so omitting it leads to incorrect coefficients

Correct move:

Write the factorial denominator explicitly for each term before substituting derivative values

Wrong move:

Making differentiation errors when deriving higher order derivatives for ODE series solutions

Why:

Differentiation errors propagate to all higher order derivatives, leading to incorrect coefficients for all higher terms

Correct move:

Check each differentiation step carefully, and verify that initial condition substitution gives consistent derivative values

Wrong move:

Conflating Maclaurin/Taylor series with the FP2 method of differences topic

Why:

These are separate topics with distinct techniques, so applying the wrong method gains no marks

Correct move:

Use derivative calculations for Maclaurin/Taylor series questions, and telescoping sum methods for method of differences questions

6. Quick Reference Cheatsheet

Expansion Type	General Formula	Use Case
Maclaurin Series	(f(x) = \sum_{r=0}^n \frac{f^{(r)}(0)}{r!}x^r + ...)	Expansions centred at (x=0), standard function expansions
Taylor Series (about (a))	(f(x) = \sum_{r=0}^n \frac{f^{(r)}(a)}{r!}(x-a)^r + ...)	Expansions centred at any point (x=a), approximations near (a)
ODE Series Solution	Compute (y^{(r)}(a)) via repeated ODE differentiation, substitute into series form	Power series solutions of ODEs with given initial conditions
Standard Expansions	Provided in formula booklet: (e^x, \sin x, \cos x, \ln(1+x), \arctan x)	Shortcut for composite function expansions when derivation is not required

7. Frequently Asked

Do I need to memorise standard Maclaurin expansions for the exam?

No, all standard expansions (e^x, sin x, cos x, ln(1+x), arctan x) and their validity ranges are provided in the official FP2 formula booklet. However, you must be able to derive these from first principles when explicitly asked in a question.

What is the difference between a Maclaurin and Taylor series?

A Maclaurin series is a special case of a Taylor series where the expansion is centred at (a=0). Taylor series can be constructed for any centre point (a).

Can I use a calculator to compute derivatives for series questions?

Basic scientific calculators are permitted, but CAS-enabled calculators are banned. You must show full working for all derivative calculations to gain full marks, even if you use a calculator to check values.

Going deeper

What's Next

Now that you have mastered Maclaurin and Taylor series for Edexcel IAL FP2, you can apply these skills to solve a wide range of exam questions, including function value approximations, composite function expansions, and series solutions for complex differential equations. This topic is often combined with other FP2 content such as polar coordinates and further calculus, so cross-topic practice is critical for exam readiness. Make sure to complete past paper questions on this topic to familiarise yourself with common phrasing and mark scheme requirements, and review the official FP2 formula booklet to confirm which expansions are provided for exam use.