Editing Big O notation (section)

=== Infinitesimal asymptotics ===
Big O can also be used to describe the [[Taylor series#Approximation error and convergence|error term]] in an approximation to a mathematical function. The most significant terms are written explicitly, and then the least-significant terms are summarized in a single big O term.  Consider, for example, the [[Exponential function#Formal definition|exponential series]] and two expressions of it that are valid when {{mvar|x}} is small:
<math display=block>\begin{align}
e^x &=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\dotsb &&\text{for all finite } x\\[4pt]
    &=1+x+\frac{x^2}{2}+O(x^3)                                &&\text{as } x\to 0\\[4pt]
    &=1+x+O(x^2)                                              &&\text{as } x\to 0
\end{align}</math>
The middle expression (the one with <math> O(x^3)</math>) means the absolute-value of the error <math> e^x-(1+x+\frac{x^2}{2})</math> is at most some constant times <math> |x^3|</math> when <math>x </math> is close enough to <math> 0</math>.