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"