Make Egyptian fractions a little more robust.

Add checking for negative arguments, and arguments > 1

Don't know if this is actually needed; it does help avoid endless loops if you give an improper fraction to some of the algorithms.  Feel free to reject.
> Is there a better way to represent these "illegal" Egyptian fractions?

I think, yes.  1/1+1/2+1/3+1/4... - harmonic series, it's divergent.  So, instead of [1,1,3] we can have [1,2,3,4,5,20] for 7/3.

> Or maybe it would be better just to reject such inputs altogether.

Yeah.  I think - most algorithms assume 0 < Rational < 1.