Order of approximation

Formal expressions

In science and engineering, it is common to approximate a function by applying a series expansion such as a Taylor series, and then discarding terms with higher powers of the variable. The order of the approximation is the highest power term that is kept. For example, one can refer to a zeroth-order approximation, a first-order approximation, a second-order approximation, etc. Variations such as zero-order approximation and order-zero approximation are also seen.

In the case of a smooth function, the nth-order approximation is a polynomial of degree n, which is obtained by truncating the Taylor series to this degree. Truncating the series affects accuracy. In most cases the accuracy of the approximation improves as the order increases, but the order does not directly indicate the percent error of the approximation. See Taylor's theorem for more on this.

For example, in the Taylor expansion of the exponential function, $${\displaystyle e^x=\underset{0^{\text{th}}}{\underbrace{1}}+\underset{1^{\text{st}}}{\underbrace{x}}+\underset{2^{\text{nd}}}{\underbrace{\frac{x^2}{2!}}}+\underset{3^{\text{rd}}}{\underbrace{\frac{x^3}{3!}}}+\underset{4^{\text{th}}}{\underbrace{\frac{x^4}{4!}}}+\dots ,}$$ the zeroth-order term is ${\displaystyle 1;}$ the first-order term is ${\displaystyle x,}$ second-order is ${\displaystyle x^2/2,}$ and so forth. If $|x|<1,$ each higher order term is smaller than the previous. If $|x|\ll 1,$ then the first-order approximation, $${\displaystyle e^x\approx 1+x,}$$ is often sufficient. But at ${\displaystyle x=1,}$ the first-order term, ${\displaystyle x,}$ is not smaller than the zeroth-order term, ${\displaystyle 1.}$ And at ${\displaystyle x=2,}$ even the second-order term, ${\displaystyle 2^3/3!=4/3,}$ is greater than the zeroth-order term.

For $|x|<1,$ the first few orders of approximation of this function are:

Zeroth-order

${\displaystyle e^x\approx 1}$

First-order

${\displaystyle e^x\approx 1+x}$

Second-order

${\displaystyle e^x\approx 1+x+\frac{x^2}{2}}$

Third-order

${\displaystyle e^x\approx 1+x+\frac{x^2}{2}+\frac{x^3}{6}}$

Fourth-order

${\displaystyle e^x\approx 1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}}$