A sequence of functions 
is an asymptotic sequence as
if ![phi_(n+1)(x)=o([phi_n(x)])](/images/equations/AsymptoticExpansion/Inline3.svg)
for every 
.
A formal series 
, where formal means that convergence
is not assumed, is an asymptotic expansion of 
if, for every positive
integer 
,
![f(x)-sum_(n=0)^(N-1)a_nphi_n(x)=o[phi_(N-1)(x)],](/images/equations/AsymptoticExpansion/NumberedEquation1.svg) |
as 
.
The successive partial sums approximate 
to successively smaller asymptotic orders.
