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<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>
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\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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