Maclaurin and Taylor Series

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Edexcel IAL Further Pure Mathematics 2 (WFM02)

Maclaurin and Taylor Series

The big idea: A smooth function can be rebuilt near a point from its value and its derivatives there. Maclaurin series use the point x = 0; Taylor series use any point a.

Summary — What This Chapter Covers

Maclaurin: f(x) = f(0) + f′(0)x + f″(0)x²/2! + f‴(0)x³/3! + …
Taylor: f(x) = f(a) + f′(a)(x − a) + f″(a)(x − a)²/2! + …
Standard series (e^x, sin x, cos x, ln(1 + x), (1 + x)ⁿ) can be combined by substituting, multiplying and differentiating.
ln(1 + x) is valid only for −1 < x ≤ 1, so ln x itself has no Maclaurin series — use a Taylor series about x = 1.
The Taylor series method solves differential equations as a power series.

1. Maclaurin Series

Maclaurin

Building a series at x = 0

Differentiate repeatedly, evaluate each derivative at 0, and substitute into the formula. Often it is quicker to combine standard series.

Standard series e^x = 1 + x + x²/2! + x³/3! + … sin x = x − x³/3! + x⁵/5! − … cos x = 1 − x²/2! + x⁴/4! − … ln(1 + x) = x − x²/2 + x³/3 − …

Worked Example

Find the Maclaurin series of ln(cos x) up to the term in x⁴.

Practice Question 1

Find the series for e^x sin x up to the term in x³.

2. Taylor Series

Taylor

Expanding about x = a

Use Taylor series when you need an approximation near a point other than 0, or when f(0) is undefined.

Worked Example

Expand sin x in ascending powers of (x − π) up to the term in (x − π)³.

Practice Question 2

Find the Taylor series of ln x about x = 1, up to the term in (x − 1)³.

3. Series Solutions of Differential Equations

Series solutions

Using the equation to find the derivatives

Use the differential equation and the initial conditions to find y(0), y′(0), y″(0), …, differentiating the equation itself to get higher derivatives. Then substitute into the Maclaurin (or Taylor) formula.

Worked Example

⁴ ² ²

Memorise

Maclaurin

Σ f^(r)(0)x^r/r!

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Maclaurin and Taylor Series