I am using Mathematica 8 to do lengthy asymptotic expansions for use in statistics. In particular I have


where the first term is of order $O(n^{-1/2})$, the second one is $O(n^{-1})$, the third is $O(n^{-3/2})$ an the last one is $O(n^{-2})$ (where n is some asymptotic index). What I want to do is to expand

Series[Log[1 + A*x], {x, 0, 4}]

where $x=\lambda$ and A is of order $O(n^{-1/2})$, and I want Mathematica to be able to automatically retain all the terms up to $O(n^{-2})$. I have tried to expand and to use Simplify by telling the orders of the components in $\lambda$ through Assumptions, but I am not still able to get anywhere. Does anybody knows a way to do this or a good tutorial to learn from?

Thank you in advance,



This will do the expansion in terms of eps=1/n around 0.

l = a eps^(1/2) + b eps^1 + d eps^(3/2) + g eps^2 + h eps^(5/2);
Series[Log[1 + A eps^(1/2) l], {eps, 0, 2}]
(* a A eps+A b eps^(3/2)+(-(1/2) a^2 A^2+A d) eps^2+(-a A^2 b+A g) eps^(5/2)+O[eps]^3 *)
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