Looking at the implementation, it seems that the ".series" method uses Pynac/Ginac series. A little experimentation seems to suggest that this is not properly wrapped. If we break up the expression in small parts and see how series expansions of the different components combine, we get inconsistent results:
sage: A=(1-x).series(x,6) sage: B=sqrt(1-6*x+x^2).series(x,6) sage: C=(2*x).series(x,6) This seems correct: sage: A,B (1 + (-1)*x + Order(x^6), 1 + (-3)*x + (-4)*x^2 + (-12)*x^3 + (-44)*x^4 + (-180)*x^5 + Order(x^6)) This seems correct too: sage: A.truncate()-B.truncate() 180*x^5 + 44*x^4 + 12*x^3 + 4*x^2 + 2*x This is a mess: sage: (A-B).truncate() -(1 + (-3)*x + (-4)*x^2 + (-12)*x^3 + (-44)*x^4 + (-180)*x^5 + Order(x^6)) + (1 + (-1)*x + Order(x^6)) This seems fine again: sage: (A-B).series(x,6) 2*x + 4*x^2 + 12*x^3 + 44*x^4 + 180*x^5 + Order(x^6) And this seems to be a way of getting a correct answer out (if a little lossy): sage: ((A-B)/C).series(x,6) 1 + 2*x + 6*x^2 + 22*x^3 + Order(x^4) so it seems that the series code goes haywire when removing the pole. It looks like an "upstream" bug in Pynac (and possibly Ginac). There are other problems with "series" objects:. They don't round-trip to maxima very well: sage: (1/(1-x)).series(x,10).simplify() ValueError: The name "1" is not a valid Python identifier. so it looks like either "series" needs a lot of love and support in SR, or needs to be pulled. Either one needs significant work from a sufficiently expert programmer. In the mean time, you can accomplish your computations without using SR: sage: R.<x>=QQ[[]] sage: (1 - x - sqrt(1 - 6*x + x^2))/(2*x) 1 + 2*x + 6*x^2 + 22*x^3 + 90*x^4 + 394*x^5 + 1806*x^6 + 8558*x^7 + 41586*x^8 + 206098*x^9 + 1037718*x^10 + 5293446*x^11 + 27297738*x^12 + 142078746*x^13 + 745387038*x^14 + 3937603038*x^15 + 20927156706*x^16 + 111818026018*x^17 + 600318853926*x^18 + O(x^19) which is probably why the state of "series" is so abysmal: people don't use it. -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at https://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.
