Maclaurin and Taylor series

What you’ll learn

• Build expansions from successive derivatives.
• Use series for local approximations and equations.

Understand the idea

A Taylor expansion describes a function near a chosen point through its derivatives there. A Maclaurin expansion is the special case centred at zero. The factorial in each denominator compensates for repeated differentiation.

For a composite expression, substitute into a known expansion or multiply and collect terms only up to the requested degree. A truncated series is a local approximation: state the order retained and respect the convergence interval of any infinite expansion. Differential equations can also generate successive derivatives at an initial point.

Key formulae

$$f(a+h)=f(a)+hf'(a)+\frac{h^2}{2!}f''(a)+\cdots$$

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

$$\sin x=x-\frac{x^3}{3!}+\cdots$$

$$\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots\quad(|x|<1)$$

Worked example

Expand $e^{2x}$ through the term in $x^3$.

1. Begin with $e^u=1+u+u^2/2+u^3/6+\cdots$.
2. Substitute $u=2x$.
3. Simplify each coefficient: $4/2=2$ and $8/6=4/3$.

Answer: $e^{2x}=1+2x+2x^2+\frac43x^3+\cdots$.