= Asymptotic power series
{title2=$f(x)-\sum_{k=0}^Nc_k(x-a)^k=o(|x-a|^N)$}
An <asymptotic power series> uses powers as its <asymptotic sequence>. Near a finite point $a$, the condition in the title applies along the specified domain for every fixed $N$; at <infinity>, inverse powers are convenient. It need not be a convergent <power series>. Its <coefficients> record the successive limiting responses, as the integral example below demonstrates.
The finite-order <Taylor theorem> supplies an <asymptotic power series> for a <smooth function> near a point, but <smoothness> does not imply that its <Taylor series> converges to the <function>. Define $g(x)=e^{-1/x}$ for $x>0$ and $g(x)=0$ for $x\leq0$. Every right-hand <derivative> is an <exponential function> times a <polynomial> in $1/x$, tending to zero as $x\downarrow0$. Consequently $g$ is a <smooth function> with every <derivative> zero at the origin. Its all-zero <Taylor series> misses $g(x)>0$ to the right. Zero and $g$ also have the same all-zero <asymptotic power series> on that side, despite being different <functions>. An <asymptotic expansion> can determine all its own <coefficients> without determining the represented <function> uniquely.
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