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The choice of series expansion depends on the [[scientific method]] used to investigate a [[Phenomenon#Scientific|phenomenon]]. The expression '''order of approximation''' is expected to indicate progressively more refined approximations of a [[Function_(mathematics)|function]] in a specified [[Interval_(mathematics)|interval]]. The choice of order of approximation depends on the [[Research|research purpose]]. One may wish to simplify a known [[Closed-form_expression#Analytic_expression|analytic expression]] to devise a new application or, on the contrary, try to [[Curve_fitting|fit a curve to data points]]. Higher order of approximation is not always more useful than the lower one. For example, if a quantity is constant within the whole interval, approximating it with a second-order [[Taylor series]] will not increase the accuracy.
In the case of a [[smooth function]], the ''n''th-order approximation is a [[polynomial]] of [[degree of a polynomial|degree]] ''n'', which is obtained by truncating the Taylor series to this degree. The formal usage of ''order of approximation'' corresponds to the omission of some terms of the [[Series_(mathematics)|series]] used in the [[Series_expansion|expansion]]. This affects [[Accuracy_and_precision|accuracy]]. The error usually varies within the interval. Thus the terms (''zeroth'', ''first'', ''second,'' etc.) used above meaning do not directly give information about [[percent error]] or [[significant figures]]. For example, in the [[w:Taylor's theorem|Taylor series]] expansion of the [[Exponential_function#Formal_definition|exponential function]],
<math display="block">e^x=\underbrace{1}_{0^\text{th}}+\underbrace{x}_{1^\text{st}}+\underbrace{\frac{x^2}{2!}}_{2^\text{nd}}+\underbrace{\frac{x^3}{3!}}_{3^\text{rd}} + \underbrace{\frac{x^4}{4!}}_{4^\text{th}} + \
the zeroth-order term is <math>1;</math> the first-order term is <math>x,</math> second-order is <math>x^2/2,</math> and so forth. If <math>x<1,</math> each higher order term is smaller than the previous. If <math>x</math> is very small <math>(x<<1)</math>, then the first order approximation,▼
▲<math display="block">e^x=\underbrace{1}_{0^\text{th}}+\underbrace{x}_{1^\text{st}}+\underbrace{\frac{x^2}{2!}}_{2^\text{nd}}+\underbrace{\frac{x^3}{3!}}_{3^\text{rd}} + \underbrace{\frac{x^4}{4!}}_{4^\text{th}} + \mathcal O^5 </math>
▲</math> and so forth. If <math>x<1,</math> each higher order term is smaller than the previous. If <math>x</math> is very small <math>(x<<1)</math>, then the first order approximation,
<math display="block">e^x\approx 1+x, </math>
is often sufficient. But at <math>x=1,</math> the first-order term, <math>x,</math> is not smaller than the zeroth-order term, <math>1.</math> And at <math>x=2,</math> even the second-order term, <math>2^3/3!=4/3,\,</math> is greater than the zeroth-order term.
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