> Limits are tricky. You said that maxima should find the first non-zero
> term in a taylor series. Right. What about
> taylor(sin(x)^2+cos(x)^2-1,x,0,??);
> Even for ??=1000000, maxima will return 0 + ..., after spending a lot of time...
This is really a new discussion, but...
There must be some argument that shows that you only need to calculate
the first N terms before you know whether expressions like the above
are identically zero.
Taylor has some form of that internally so that it can calculate
sufficient terms for subexpressions like taylor( (sin(x)-x)^(1/3) , x,
0, 1). This calculation isn't perfect -- try for example
taylor(sqrt(sin(x)^2+cos(x)^2-1),x,0,2) and get 'taylordepth exceeded
while expanding'.
I wonder if we can refine it. If so, it would give us a nice
technique for proving things to be identically zero.
-s