[Newbie:] How to discard higher orders of variables?

My suggestion may not be that different from the other suggestions, 
which I must admit to not having studied carefully, but here goes:

let b: taylor(beta,beta,0,2);
then you can use expressions like

sqrt(1+b^2)  and have the taylor series expanded just as you like.

My own experience in trying to reproduce hand calculations is that if 
they are quite bulky, the authors have been clever in reducing the 
amount of algebra.  Using a computer you have a choice, sometimes, of 
trying to duplicate the cleverness by nice programs, and just using 
brute force.  Sometimes the brute force cannot do the job, but sometimes 
it can!

I was finding expansions for the onset of turbulence in a fluid flow 
(Taylor vortex).

Have fun!
RJF