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I'll give a concrete example and I hope that my general question will be clear.

Say I have three variables, $f$, $g$, $h$, and I know that $f=\mathcal O(x)$, $g=\mathcal O(x^2)$, $h=\mathcal O(x^3)$ ($\mathcal O$-notation) or something of that sort. Now, I want to do a series expansion of expressions involving $f$, $g$, $h$ to a consistent order in $x$. For example, if I write Series[..., {x,0,4}] I want to keep terms like f^3 and f g but to throw away terms likef g^2 and g h.

Is there a simple way to do so?

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up vote 11 down vote accepted

Try this

Normal@Series[expr/.{f->x f, g->x^2 g, h->x^3 h},{x,0,4}] /.x->1

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