#1084 deftaylor

open
nobody
5
2013-06-08
2007-01-22
Barton Willis
No

deftaylor works OK for expansion at zero, but don't
try to expand anywhere else:

(%i1) deftaylor(f(x), sum(x^k,k,0,inf));
(%o1) [f]
(%i2) taylor(f(x),x,0,2);
(%o2) 1+x+x^2+...
(%i3) taylor(f(x),x,1,2);
`taylor' unable to expand at a point specified in:
sum(x^*index,*index,0,inf)
-- an error. Quitting. To debug this try debugmode(true);

Notice the garabage in the error message. For
an expansion at a nonzero number, it would
be better if Maxima used the differentiate and
evaluate scheme instead of trying to use the
deftaylor info.

Discussion

  • Chip Eastham
    Chip Eastham
    2013-06-08

    Since sum(x^k,k,0,inf) is singular (a simple pole) precisely at x=1, it seems unlikely that deftaylor or taylor should succeed in providing a series expansion there.

    A better example of the failure, which can be demonstrated with taylor only (not deftaylor), would be expansion at x=1/2:

    taylor(sum(x^k,k,0,inf),x,1/2,15);

    taylor: unable to expand at a point specified in: (etc.)

    If the limit of summation inf is changed to a finite value, e.g. 30, then the above expansion succeeds. However expansion around x=1 also succeeds in this case (as the infinite sum has been demoted to a high degree polynomial).

    Maxima version: "5.28.0-2"
    Maxima build date: "2012-08-27 23:16:48"
    Host type: "i686-pc-mingw32"
    Lisp implementation type: "GNU Common Lisp (GCL)"
    Lisp implementation version: "GCL 2.6.8"