The Maclaurin series of a function, the standard series for e^x, ln(1+x), sin x and cos x, finding a series by repeated differentiation or by combining known series, and using a truncated series to approximate values.

What this dot point is asking

OCR wants you to construct the Maclaurin series of a function, either by repeated differentiation and the general formula or by substituting into and combining the standard series, to quote the standard expansions for $e^x$, $\ln(1 + x)$, $\sin x$ and $\cos x$, and to use a truncated (finite) series to approximate a function value.

The Maclaurin formula

The Maclaurin series is the Taylor series about $x = 0$. Each coefficient is a derivative evaluated at zero, divided by a factorial. It represents a function near the origin as a polynomial of (in principle) infinitely many terms.

The standard series

These four expansions are quoted constantly and are given in the OCR formulae booklet; knowing them by heart saves repeated differentiation.

Finding a series by repeated differentiation

When no standard series fits directly, differentiate the function repeatedly, evaluate each derivative at $x = 0$, and assemble the coefficients. This is the safe method for products such as $e^x\cos x$ or for an unfamiliar function.

Finding a series by combining known series

Often the fastest route is to substitute into a standard series or to add, subtract or multiply known series. Substituting $-x^2$ into the $e^x$ series gives the series for $e^{-x^2}$ instantly; multiplying the $\sin x$ and $\cos x$ series term by term gives the series for $\sin x\cos x$.

Using a series to approximate

Truncating a Maclaurin series after a few terms gives a polynomial approximation, accurate when $x$ is small because the omitted terms involve higher powers of a small number. The more terms you keep, the better the approximation near the origin.

The Maclaurin series links differentiation to series and feeds into approximation and into the further-calculus integration techniques.

Try this

Q1. Write the Maclaurin series of $\cos 2x$ up to the term in $x^2$. [2 marks]

• Cue. Substitute $2x$ into $\cos u = 1 - \tfrac{u^2}{2} + \cdots$: $1 - \tfrac{(2x)^2}{2} = 1 - 2x^2$.

Q2. Use the first two non-zero terms of the series for $\sin x$ to estimate $\sin 0.2$. [2 marks]

• Cue. $\sin x \approx x - \tfrac{x^3}{6}$, so $\sin 0.2 \approx 0.2 - \tfrac{0.008}{6} \approx 0.19867$.