An asymptotic series is a series expansion of a function in a variable which may converge
or diverge (Erdélyi 1987, p. 1), but whose partial sums can be made an
arbitrarily good approximation to a given function for large enough . To form an asymptotic
series of
(1)
take
(2)
where
(3)
The asymptotic series is defined to have the properties
(4)
(5)
Therefore,
(6)
in the limit . If a function has an asymptotic
expansion, the expansion is unique. The symbol is also used
to mean directly similar.
Asymptotic series can be computed by doing the change of variable and doing
a series expansion about zero. Many mathematical operations can be performed on asymptotic
series. For example, asymptotic series can be added, subtracted, multiplied, divided
(as long as the constant term of the divisor is nonzero), and exponentiated, and
the results are also asymptotic series (Gradshteyn and Ryzhik 2000, p. 20).
which may converge
or diverge (Erdélyi 1987, p. 1), but whose partial sums can be made an
arbitrarily good approximation to a given function for large enough 
. To form an asymptotic
series 
of
![x^nR_n(x)=x^n[f(x)-S_n(x)],](/images/equations/AsymptoticSeries/NumberedEquation2.svg)
. If a function has an asymptotic
expansion, the expansion is unique. The symbol 
is also used
to mean directly similar.
and doing
a series expansion about zero. Many mathematical operations can be performed on asymptotic
series. For example, asymptotic series can be added, subtracted, multiplied, divided
(as long as the constant term of the divisor is nonzero), and exponentiated, and
the results are also asymptotic series (Gradshteyn and Ryzhik 2000, p. 20).
